In U.S. English, using the "register" (degree of formality) appropriate for a written advertisement, I would say Our approach is different from our competitors'. I think that's perfectly correct--as long as you put in the apostrophe! (It's short for "Our approach is different from our competitors' approach," and "competitors'" is a possessive adjective). Our approach is different from that of our competitors. Here, you don't use the apostrophe because we are just using the noun "competitors," not the possessive adjective. In real life, though, I think you might shorten it even more and just say: Our approach is different.

It sounds to me like "whose" should be "those which" in this sentence - "those" indicating something that isn't a part of what you're describing (for example, your methods, instead of someone else's), and "which" being more appropriate for narrowing that describing a non-living "thing" (e.x. a method used by your competitors) while "who" is more appropriate if you're describing people (e.x. your competitors themselves).

"Our approach is different to *those which* our competitors use" is more correct. You could also use *those that* too. The use of "whose" is incorrect like you rightly guessed.