Does there exist a finite automaton for the given language?
Your reasoning about comparisons would be correct, if there were infinitely many numbers. But there's only three of them modulo $3$.

The automaton can be constructed directly. You need three states, one for each of the residues: $0$, $1$ and $2$. Then every $a$-transition should correspond to an increment and $b$-transition to a decrement.