And no, we’re not talking about Nielsen ratings. 
Let’s make it pretty simple. House and Veronica have to switch places - Veronica goes to the hospital, has to figure out a tough mystery disease. House goes to Neptune and has to take on a PI case.
First one to the CORRECT solution wins.
What would happen? 
And, as a bonus question - who can write the funniest flashback scene to describe how they made the bet in the first place? 
House would solve the mystery fairly quickly because he apparently has some deductive reasoning skills, and Veronica would not solve the medical problem because Veronica has no medical training of any kind and is completely unprepared to solve such a question.
I agree with Otto, Veronica has no medical training whatsoever and would be lost trying to solve a medical case. Besides, she wouldn’t be allowed to see patients or their files since she doesen’t have a medical license.
Supposing House did end up in Neptune he wouldn’t give rats ass about anything going on there outside of a hospital. Though he might get arrested by Sheriff Mars on possession or attempt to purchase charges if he has to stay in Neptune for too long.
BTW, in my house Veronica Mars won the Tuesday night slot war because my wife controls the remote. Thank gods for Tivo.
Veronica would wheedle the correct diagnosis out of some passing doctor. House, meanwhile, would make several half-assed guesses at the mystery, all wrong, then pull the correct answer out of his ass in the last five minutes of the show.
I predict that Veronica will be the next bride of Wilson. She’ll sweettalk him into diagnosing the mystery ailment long before House is able to harangue the residents of Neptune into giving him any useful information.
House hands down. Veronica knows nothing about medicine, while House knows everything about the dark side of human nature.
I think ya’ll are missing the premis of the bet. It’s not that Veronica is going to try to solve some medical issue…it’s that she’s at the hospital trying to solve one of her regular-type “mysteries” and House is somehow in Neptune trying to diagnose a medical issue. Characters switch locations in other words, not specialties.
I’m going to have to go with House on this one. Veronica derives a lot of her solutions via the support of the ‘network’ she has in place in her home turf. She does a lot by contacting people she knows, pulling in favors, or running scams to get information in environments in which she is familiar.
House on the other hand pretty much takes all his medical knowledge with him. He’s already solved a medical emergency on a plane, and another one remotely from a hotel. So my bet is with him finding the solution first.
Edited to add Whoops…re-read the OP, and ya’ll were right, I was wrong. The answer is still House, but for different reasons. Veronica lacks any sort of background that would give her even a chance…whereas House has demonstrated that he’s a keen observer and can deduce much from those observations.