Sometimes I see proofs where the induction case is stating that the statement is true for all $x$ smaller than $n$, and that it needs to be proven for $n$. And sometimes it is true for $n-1$, and needs to be proven for $n$.
What is the difference between these two cases? How to know which one to use when proving?
On
The first one you mentioned is known as 'Strong induction' while the second one is 'Induction' or 'Weak induction'.
Strong and weak induction are equivalent techniques. You can use either to prove a statement, whichever one is more convenient.
The difference is that Strong induction allows you to assume more, giving you more weapons initially to prove the statement.
To know which one to use, you need to analyse the question. Usually, people begin with weak induction. If it does not seem to work and you feel like you need more than just the $n-1$ case then use strong induction.
Strong induction works sort of like regular/normal/weak induction. In regular induction, we suppose the given statement (let's call it $P(k)$) to be true, and prove this forces $P(k+1)$ to be true as well. But in strong induction, we assume that all the statements $P(1),$ $P(2)$, $\ldots$, $P(k)$ are true and prove they force $P(k+1)$ to be true as well.
To be blunt, there isn't an easy way to tell what kind of induction you should try. You just have to experiment. It's kind of like trying to see what series test would work to prove if a series converges or diverges; you'd have to try a lot of series tests to see which one works. In my experience, strong induction was useful when I found out that supposing $P(k)$ as true in the inductive step of regular induction wouldn't easily imply $P(k+1)$ to also be true.
I mention an example here in this post: Strong Induction vs Weak Induction.
Hopefully my answer helps.