My experience is with US admissions in Computer Science at a productive albeit mid-tier graduate program. Both tests were required (TOEFL only for English as a second language speakers) and both tests are used for reasons to disqualify someone rather than admit them to the program.
My grad department would get about 500 applications for about 15 slots per year. About half of those were from totally unknown schools in China or India and were rejected out of hand, just because we had no idea about the quality of those students. Of the 250 remaining, about half of those had substantially deficient transcripts- things like missing or woeful TOEFL and GRE scores. The remaining roughly 120 applications were looked at by the graduate admissions committee, and then they picked about 30 high quality candidates and circulated those applications among faculty who were known to be looking for new students.
You'll notice there are two distinct types of decisions there. The first two phases were looking for reasons to disqualify people. The third phase was looking for good reasons to pass people along to final consideration. My guess is that virtually all departments would only use TOEFL and GRE scores in this way.
In other words, a bad GRE score can hurt your chances, but there's little difference between an acceptable GRE score and a great GRE score.
TOEFL might be a little different- our experience is that even people with good TOEFL scores might be atrocious language speakers, so our department always arranged in-person interviews via Skype or similar as a way to assess language proficiency. A great TOEFL score definitely didn't mean anything other than an acceptable score, because the in-person interview was the actual test of proficiency. I will note that we had much more stringent language expectations of Asian students than those from Europe.
Another thing to consider- I was told flat out by the University graduate admissions department that I would not be accepted with my GRE score, which was 20 or 30 points below their guidelines. The actual department I applied to didn't care. They were far more impressed by my independent study and selection of a challenging senior project than they were concerned about GRE scores.