I participated in the International Math Olympiad,
and have a PhD in operations research,
which is essentially a type of applied math.
There is some overlap between math olympiads and research math.
However, as others have noted, mathematics is a very broad field,
which includes subfields such as:
algebraic topology, theoretical computer science, combinatorics,
control theory, optimization, statistics/machine learning.
The amount of overlap with math olympiads depends very much on
what subfield of "research math" you are referring to.
Math olympiads have more overlap with say combinatorics,
and less with say control theory.
In training for math olympiads,
I learned how to try special cases
to get intuition about how a problem works,
and how to simplify a problem step by step,
and how to write a logical and complete proof.
I also learned how to persevere,
and to enjoy the challenge of tackling problems which are difficult,
and also when to give up when sometimes I am just stuck.
I think that these are skills that also are required in research math.
I think that there is positive but imperfect correlation
between performance at math olympiads
and performance in research math.
If you don't do very well at the IMO,
you can still be a successful pure math professor;
and if you get a perfect score at the IMO,
that does not mean you are guaranteed
to have a successful research math career.
Finally, the math olympiad is an artificial competition,
in that the problems in the olympiad
can all be solved in a fairly short time
with a relatively small set of tricks.
On the other hand, in the real world,
research math is much more open ended,
you need to find and define your own research problems,
and oftentimes the problems cannot be solved!
I would use the analogy that
olympiad math is like an RPG such as
The Legend of Zelda: Breath of the Wild,
whereas research math is like real life,
extremely messy and open ended.