Battles are free, head-to-head competitions. You're ranked against other players, and the top finishers win a share of the Purse.
Challenges are something you purchase. Each Challenge includes a Combine Entry, a Combine Bankroll, and Tuition Points. You're measured against targets rather than other players, and passing the Combine earns you free access to the Pro Levels, where you can earn Performance Bonuses.
The two are tracked separately. Battle results don't carry over to Challenges, and Challenge results don't affect your Battle rankings.