My bank has recently changed me from Visa to Mastercard after 15 years with Visa. Presumably Mastercard are now offering them better commissions!
Anyway, to answer the question, fundamentally there's no difference, especially if they are both offering the same rates and the same credit limits. Theoretically it's possible you could find a retailer who takes one card but not the other though in practice I've never seen one in the UK who differentiates. Possibly there are some foreign countries where one works but the other doesn't though. Not any I've visited but there are probably some out there.
The only place I've ever had a difference is with UEFA, the European football people. They are sponsored by Mastercard and, as a result, ONLY take Mastercard as a credit card option. You cannot use Visa with them which was a right pain in the backside until recently for me as any ticket applications couldn't be made on my Visa. Now I have a Mastercard it's no longer an issue.