What is the difference between "infinitely many $n$" and "large enough $n$"? I'm faced with this sentence in a solution of a problem:

The negation of "$P(n)$ for infinitely many $n$" is "$\lnot p(n)$ for large enough $n$".

What is the difference between infinitely many and large enough $n$?

When something is true for infinitely many $n$, it doesn't have to be true always even when $n$ gets massive. For example, $P(n)$ might be "$n$ is a prime". This statement is true for infinitely many $n$, but it's not the case that as soon as $n>n_0$ for some number $n_0$, every $n$ is an prime number.
On the other hand, if something is true when $n$ is large enough, then it's true for infinitely many $n$. This is because there are infinitely many numbers.
