I can think of a number of things this may do for your test results.
The obvious: consider the situation where your product isn't yet available, or you're developing a new feature. Learning from mistakes others already made, or from what they did right, is immensely valuable.
It may also help your judgement of the problems the test reveals. I think we all sometimes fall into the trap of downplaying the shortcomings of something we created, while we have no trouble being quite critical of the work of others. In testing your competitor's products you may find shortcomings you were overlooking in your own product. You'll look at it with fresh eyes and it can provide a reality check looking at your own product (again) can't. Especially if your doing an expert/heuristics review this could be very valuable.
Users will have learned certain problem solving tactics that they'll apply without being conscious about doing so. They might be very handy doing something on a competitor's site, but if that tactic is not available in yours it will not appear in your test. They might tell you something about your product isn't very handy, but they probably won't be able to tell you what. So, looking at people using your competitor's product might reveal such tactics.
Same goes for features in a competitor's product you may not have thought were important. Users may not tell you they use them, but tests might reveal that they're important anyway. Note that this means that the fact that your product and that of the competitor are not the same could be valuable too.
Something else: you're inviting people to judge your product. Of course you'll prepare them carefully, but they may be holding back criticism. Looking at a competitor's product could solicit feedback you were not getting on your own product.
Although this does not strictly relate to usability testing, I've found that in context mapping sessions anecdotes and comments that happen around the actual session can be very informative. Unless your testing for a specific problem area, this to me means that breaking away from the core test can give you a lot of useful input for the (interaction) design of the product.
There's probably more.