Peter Flom gave a good answer but I want to address some other aspects. Most of this answer will focus on pure math, but aspects will apply to other STEM fields. (It is also worth noting that a lot of amateurs do good work, or do fun work in things like recreational math and we don't notice them quietly being productive.)
One: there are certain classes of mistakes that when you get a regular education in a topic get pointed out to you, as well as theorems which show that a particularly approach to something simply cannot work. Here are three examples in pure math which amateurs seem to miss: First, the relativization barrier and natural proofs barriers showing that a lot of strategies for showing that P != NP cannot work(relativization also blocks a lot of strategies for showing that P=NP). Second, the orbit of -1 shows that a lot of naive modular arithmetic approaches to the Collatz conjecture must fail. Third, the existence of Descartes number shows that a lot of strategies for proving that no odd perfect numbers exist, while not actually using the genuine primality of the prime factors must fail.
Edit: I'm including here at the suggestion of the comment below a bit more about how these roughly work: If you don't care about this feel free to scroll down past the next few spoilered paragraphs and skip to the paragraph which starts "Curiously" and you'll get back to the general points.
For the first example, whether P=NP, the question is roughly speaking whether the set of problems which are easy to tell what the answer is is the same as the set of problems where given an answer one can quickly check this. The guess is that this is no. Roughly speaking, there should be classes of puzzles where finding the solutions are hard but checking a solution is easy. For example, solving a jigsaw puzzle feels much harder than checking that a solved puzzle is solved. Similarly, solving a Sudoku is tough, but verifying a solved Sudoku is easy. Proving this rigorously though is apparently beyond current math, but P roughly corresponds to the set of puzzles which are easy and NP corresponds to the set of puzzles which you can check solutions easily. So the conjecture that P is not equal to NP is the conjecture that there are classes of problems where it really is tough to find solutions, but checking a solution given to you is easy.
The difficulty mentioned above is that you can imagine what we call an "Oracle" which is a special machine which can answer some very tough set of questions for you, and you can then imagine puzzles where one is allowed to solve things with the Oracle as an assistant, and depending on your Oracle choice, you get different versions of P and NP. It turns out that depending on your choice of Oracle, you can get your versions of P and NP to equal each other or not be equal. But and here's key point, most of the obvious techniques to try to resolve a question like this will not look any different if there is also an Oracle attached. So that means those techniques cannot resolve this problem by themselves, since if they did, it would tell us that that occurred regardless of our Oracle choice, which is not the case.
The Collatz conjecture (sometimes called the 3n+1 problem) is one of the most famous open problems, and I should warn anyone that if you have not seen it, it is a genuinely dangerous thing to know about. You may try to spend a lot of time thinking about it once you hear about it! You've been warned. Ok: So here's the idea. We're going to start with a natural number, and when it is even, we'll divide by 2, and when it is odd, we'll multiply by 3 and then add 1. We'll keep repeating this but we'll stop if we hit 1. For example, let's say we start with 22 22 is even so we divide by 2 to get 11. 11 is odd so we multiply by 3 and add 1 to get 34. 34 is even so we divide by 2. So we get a sequence. If we write this out for 22 we get 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. The conjecture is that no matter what number we start with, we'll end up eventually hitting 1 and stopping. This seems really simple but we cannot prove it!
Now, one way of thinking about numbers that is helpful is what is called modular arithmetic. You've seen a version of this already but might not realize it. When you tell time, you use it. For example, if it is 10 o'clock, and you want to know what time it will be in 4 hours, you add 4, get 14, and then subtract 12 to conclude it will be 2 PM. So all you cared about was the remainder when you divide by 12. If you are in a country where you use 24 hour times, you'll care the same about the remainder when you divide by 24. And you do something similar with days of the week. If today is Friday, and you want to know what day it will be in 10 days, you don't need to count out 10 days. Instead, you count out 3 days, because 7 of them will loop. Modular arithmetic is looking at the remainder when you divide by a specific number this way, and is one of the most fundamental techniques of number theory. If you've seen puzzles that involve looking at just the 1s digit of a number, you are really doing modular arithmetic with respect to the number 10. (We would say the first example is mod 12, the days of the week are mod 7, the digits are mod 10, and so on).
One might hope that all one needs to do is something using some specific choice of mod to understand the Collatz conjecture. And for some very similar problems you can do that. But for the Collatz conjecture, it turns out that you cannot. Here is one way to see the barrier with a specific example; we'll look at mod 5. Suppose that we have a number n which leaves a remainder of 4 when we divide by 5, (say 9) and suppose we apply the Collatz function to it. So we multiply by 3 and add +1. Then we have that 3n+1 is leaves a remainder of 3 when you divide by 5. But then if we divide by 2, we are left with a number that is 4 mod 5 again. So we cannot use mod 5 to conclude much because we might get stuck in an indefinite sequence of numbers like this and mod 5 won't be able to notice. Something similar happens for say numbers which leave a remainder of 9 when you divide by 10, or numbers which are 55 when you divide by 56. In general, for any mod m, m-1 will act in a similar way. So this means that you cannot solve Collatz by just thinking about modular arithmetic.
Our third example: A famous old problem is whether there are any odd perfect numbers. A number is said to be perfect if when you add up all the factors of a number which are less than the number you get the number itself. For example, the factors of 6 which are less than 6 are 1, 2, and 3, and 1+2+3=6. So 6 is perfect. (In contrast, the factors of 8 which are less than 8 are 1, 2, and 4, and 1+2+4=7, so 8 is not perfect. (Note: Some people use factor to include negative factors so -2 would be a factor of 8. For simplicity I'm using factor here to mean positive factors.) The first few perfect numbers are 6, 28, 496, 8128... and they have been studied since the ancient Greeks. All known perfect numbers are even. Two of the oldest unsolved problems in all of math are are there infinitely many perfect numbers which are even, and are there any perfect numbers which are odd? Now, it turns out that for many purposes, it is easier to instead of looking at the sum of the factors of a number which are less than a number, but to include all of the factors (that is include the number itself). This leads to what is called the sigma function, denoted not too surprisingly with the Greek letter sigma, 𝜎(n). For example, 𝜎(6)=1+2+3+6=12. Note that using this function, n being perfect is equivalent to 𝜎(n)=2n. 𝜎 has a particular nice formula in terms of its prime factors. Here's an example of how it works: Take 60. It turns out that 𝜎(60)=168. Also 60 = 4 times 3 times 5, and 𝜎(4)𝜎(3)𝜎(5)=(1+2+4)(1+3)(1+5)=168. This is not a coincidence! This works in general as long as one keeps powers of different primes. Note they really have to be different primes. It is not true for example that 𝜎(8)=𝜎(2)𝜎(4).
Now, Descartes noticed that the number D=198585576189 had a very interesting property. D= (3^2)(7^2)(11^2)(13^2)(22021). So if one takes 𝜎(D) one has 𝜎(D)=(3^2)𝜎(7^2)𝜎(11^2)𝜎(13^2)𝜎(22021)= (3^2+3+1)(7^2+7+1)(11^2+11+1)(13^2+13+1)(22021+1)=2D, so D is an odd perfect number! But there's a problem which Descartes was aware of: we've calculated 𝜎(22021) wrong by assuming 22021 is prime. In fact, 22021= 19^2 times 61. Bummer. But here's the annoying thing: the calculation above almost works, except for this little problem. And a lot of proof strategies which would prove no odd perfect numbers exist would also show that no numbers which are perfect if we "forget" that a factor like 22021 is not actually prime should be able to exist either.
Curiously, all of these are also things which one sees occasional professionals do also, but often people who are from slightly adjacent areas who try to move into one of these problems.
There's also a lot of rules about how to approach things that will make one less likely to make mistakes that amateurs don't pick up on. For example, proofs by contradiction are known to be highly perilous because an algebraic mistake can simply lead to an apparent contradiction. Thus, professionals try to prove as much as they can directly in a series of lemmas, and only reserve contradictions when they put those together. This also has the advantage that one can then often check those lemmas against concrete examples. One sees similar issues in other areas; physicists for example know they need to be really careful when doing a coordinate transformation in Special Relativity.
There is an unfortunate additional issue which is ego. A lot of the amateurs have massive egos and think they are therefore the brilliant people who are going to solve major things. In fact, a lot of people have that level of ego in undergrad or early grad school. But the academic process manages to disabuse them of that to some extent, while also getting them to calibrate what problems they can work on. Amateurs have not gone through that process.
Another aspect is that often the amateurs aren't aware of the minor problems, so they spend their time beating their heads against the walls on the major or famous ones since those are the only ones they know about. For example, there is an excellent book, Richard Guy's "Unsolved Problems in Number Theory" which lists a few hundred open problems with references. About 3% of those are somewhat famous problems (Collatz, Goldbach's conjecture, twin prime conjecture, odd perfect numbers, etc.), but amateurs are often not looking at books like that. So they are not even aware of all these other worthwhile problems which have had many fewer people think about and therefore are much more likely to have low hanging fruit.
