- Reduces the power of Fire type attacks by 50%
- Lasts the entire battle
- Ends after 5 turns, or 8 if the user is holding a Damp Rock
- Power of Water type attacks increases by 50%
- Power of Fire type attacks decreases by 50%
- The accuracy of Thunder and Hurricane rises to 100%
- Pokemon with the ability Dry Skin recover 1/8 of their HP each turn
- Pokemon with the ability Swift Swim double their speed
- The move Weather Ball changes to Water type and doubles in power
- The moves Synthesis, Morning Sun, and Moonlight recover 1/4 of the user's HP, instead of 1/8 normally
As you can see, Rain Dance is infinitely more useful, as it has several additional effects that can benefit all kinds of Pokemon. Water Sport is basically useless and will never warrant wasting a turn and moveslot to use it. The only time it would make sense to use Water Sport over Rain Dance on a team is if there aren't any Pokemon that benefit from Rain Dance's additional effects and there are Pokemon with Synthesis, Morning Sun or Moonlight, but it still wouldn't serve any purpose, unless you're running a mono weak to Fire team I guess.
TL;DR: Don't use Water Sport. Ever.