What are primary uses and stat advantages of each character?

The system is actually quite complex, so if you want a full understanding of it, I'm just going to refer you to TFergusson's Party Mechanics Guide, section 1.4 has all the details.

If you just want a general idea though, here's a table that I pulled out of that guide, summing up each character's "grade" in each stat. The numbers range from 0-29, and lower numbers represent larger gains.

             Str   Vit   Mag   Spr   Dex
=================================================
     Cloud     1     6     3     4    26
    Barret     5     2    18    14    29
      Tifa     6    18    16     9    25
     Aeris    23    20     0     1    28
       Red    12    11    13     9    23
    Yuffie    16    19    11    10    24
 Cait Sith    19    22     6     4    28
   Vincent    21    22     6     4    28
       Cid    11     7    17    15    27
=================================================

Here is what I remember off of the top of my head:

 Cloud - Balanced, good at both
 Barret - Physical
 Tifa - Physical
 Aeris - Magical
 Red 13 - Balanced (His magic is quite good, don't overlook that)
 Cait Sith - Magical (pretty bad overall though)
 Cid - Physical
 Yuffie - Physical
 Vincent Valentine - Magical

I don't quite remember the affinity in terms of stats, but I do remember Yuffie always had the best luck.

If you want to grind, there are ways to get sources near the end game to boost everyone up to max stats so at that point it doesn't really matter.