There isn't one definitive answer for which is the best Overdrive mode since it's dependent on the character and who you're fighting.
For grinding out regular enemies that you kill pretty quickly, Warrior or Slayer is probably your best choice since you'll be doing most of the damage.
For enemies that hit you with more powerful attacks, Stoic or Comrade might be your best option for your damaging characters whereas Healer could be used for Yuna.
I would avoid using Ally since it charges very slow. I would also avoid Coward, Hero, Daredevil, Tactician, Avenger, Sufferer, Victor, Victim, Solo, Dancer, or Rook since none of these are very well suited to grinding.
You can find a complete list of all the different modes here along with how to make use of each mode.
The only gameplay difference is that winning awards a strength sphere. Strength spheres can be added to the sphere grid (in place of an empty location), which characters can then activate for +4 strength.
While strength spheres are rare, they are by no means unique, as they can eventually be acquired by killing Juggernaut in the Monster Arena, allowing you to obtain arbitrarily large numbers with enough time investment.
Best Answer
After 2 years, I would assume that somebody somewhere would have generated a list comparing the original vs remastered track. Unfortunately I looked and was unsuccessful
I wrote a little script to compare the track lists on Final Fantasy X HD Remaster Original Soundtrack and Final Fantasy X Original Soundtrack and I have generate a list comparing the Original OST and the Remastered OST