Toph would win. Why? Because she would know exactly what Bumi was doing, and Toph is the single best equipped character to defeat an earthbender.

Toph???s "vision" is through earthbending, so she sees the ground and all things in contact with it. She would know exactly what move Bumi was performing and with how much power the moment he started it. Also Toph is largely self thought, hence the different style of martial arts her form is based on (Southern Praying Mantis vs. the normal Hung Gar). Therefore Bumi would have no experience fighting her style, in which case his years of experience might prove a hindrance rather than an asset. On Toph side, she has plenty of experience fighting the best Hung Gar style earthbenders in existence with her tournament fighting. Bumi might be able to defeat her, but it would have to be some kind of "Mad Genius" trick (like flooding the area where the fight takes place with water) rather than though straight earthbending skill.

Also, any experienced fighter will tell you that the two hardest opponent???s to defend against are those who have more training than you (larger repertoire, more experience) and those who have no training at all (since it is impossible to predict what they are going to do, and therefore mount the right defense). Toph falls into both of these categories, in a way. She does not perform the expected moves (Hung Gar), and she has trained herself extensively.

Finally consider that in a fight, sometimes it comes down to luck; a single lucky hit, while not likely to end a fight, could change its course.