How many permutations are there for the letters in the word "meеt"?

This is not a matter of mathematics. You already gave the correct answer to your own question: It depends if you treat the two $e$'s as identical (then you have $12$ permutations) or as distinguishable (in this case you have $24$ permutations). Mathematics can't help you to decide which one fits for you.

If the word is written on a computer, using any standard text editor, then both $e$'s are identical, even if you look deep into the electronic storage. This gives you $12$ permutations.

But if you hand out colored pens to a child, ask him or her to write the word on a piece of paper, and then cut out the four letters and shift them around on your table, you will easily be able to distinguish the two versions of the letter $e$. So now you will find $24$ permutations.

You can think like this:

If the letters would be different then you would have $4!$ as the number of permutations, but in the word $MEET$ you have two letters that are indistinguishable so swapping these two doesn't give rise to a new permutation. So we have to "remove" these, and the number of ways $2$ letters can be permutated is $2!$ so we have to divide with $2!$.

Hope this helps :)

The word "meat" will have $^4P_4=24$ permutations. Consider two of these $24$ permutations, i.e
$p_1 = $ "mtae" and $p_2 = $"mtea", if we replace 'a' with 'e' in both $p_1$ and $p_2$ then they will become identical i.e "mtee". Thus two distinct permutation of the word "meat" corresponds to same permutation upon replacing 'a' with 'e'. Hence we need to divide $24$ by $2$ to get the right result. If $n$ letters are identical then the internal permutation of these $n$ letters corresponds to identical permutation of the whole word and we must then divide by $n!$.