Both possess fire-based abilities and both are more-or-less immune to the effects of fire. However, without thinking too deeply about it, Natsu would easily win this match up most of the time:
Natsu has full control over his powers. Rin doesn't.
Natsu can consume fire to replenish his energy and strength. Rin can't do anything like that.
Natsu has years of combat experience. Rin doesn't.
Natsu has several power-ups (Lightning-Flame Dragon Mode and Dragon Force). Rin doesn't.
Natsu is physically monstrous, even without using his Dragon Slayer Magic. Rin is strong, but not to the same extent, and he isn't much without his sword, Kurikara.

Honestly I think Natsu would win.