At this point, the two languages are practically identical except for syntax. You sometimes get one MS language team offering a feature in their language and then the other team offers a similar feature in the other language's next release, so they're not always exactly the same, but they can do pretty much the about same things. (I do a lot of programming with both
VB and C# and often convert examples from one language to the other and it's rarely difficult.)
There are still a few differences in where you can use them. For example, Windows Phone 7 programming has been available in C# but not
VB, but that has changed recently. Microsoft tries to minimize that sort of thing.
As for ASP.NET in particular, again I don't think there's a big difference. (Although I haven't done ASP.NET lately.)
I would just use:
- The one you feel more comfortable with, or
- The one your programming team is using, or
- The one you want to learn more, or
- The one you want to add to your resume

I think it's more a matter of personal taste that technical merit.