There's no difference. It's a matter of style.
I work with a company called IndiaNIC and we write both - a web designing and a website designing.
So, they mean the same. I think the term 'web designing' had been coined for the sake of brevity. That is, in many campaigns, we run out of space (for example, say Pay Per Click) wherein you need to restrict the number of letters. Said that, we fit 'web designing' instead of 'website designing' saving four precious letter spaces!
Web is World Wide Web and websites began with www (World Wide Web). So, in this context, web is website. And for almost anything on web, it has to have an address, which we call a 'website'.
For the World Wide Web, when you want to have a website, you search for web designer or website designer