In how many ways can five letters be posted in 4 boxes? Question : In how many ways can 5 letters be posted in 4 boxes?
Answer 1: We take a letter. It can be posted in any of the 4 boxes. Similarly, next letter can be posted in 4 ways, and so on. Total number = $4^5$.
Answer 2: Among the total number of positions of 5 letters and 3 dividers (between 4 boxes), positions of 3 dividers can selected in $\binom{8}{3}$ ways.
Which one is correct (I think the first one)? Why is the other one wrong and how does it differ from the right answer in logical/combinatorial  perspective?

The way to understand this problem (and decide which answer is correct) is to ask what you are counting exactly.  If all the letters are distinct then the first answer sounds better because it counts arrangements that only differ by where each particular letter goes. 
The second answer appears to make the letters and the dividers distinct, as well as counting as distinct the cases where two letters appear in a box in a different order.  These seem to be counting different things.
If you modify the second answer to place 3 identical dividers in a row of 5 identical letters, (choose 3 from 6 ‘gaps’), then it works fine. But those ‘identical’s are not correct (are they?).
If we tried to ‘correct’ the second solution by multiplying by rearrangements (to take into account all possible rearrangements of letters, say) we would discover that some rearrangements would be introduced that weren’t supposed to be distinct.  No, the second approach is wrong-headed: it counts the wrong things and it gets very complicated when you try to correct it.
So the answer is “the first solution is correct”.  This applies when the letters are distinct and the boxes are distinct.
