Neanderthals existed in Europe long before modern humans arrived. We both evolved from a common ancestor, thought to be Homo heidelbergensis. It follows that, if Neanderthals also wore clothes, clothes were invented more than once and the Neanderthals invented them before we did.
The two hominin species seem to have had different approaches to clothing.
In a study published in 2012, Wales estimated that Neanderthals must have covered 70-80% of their bodies during the winter months, in order to successfully live in some of the climates we know they inhabited.
Modern humans, on the other hand, needed to cover themselves up slightly more, up to 90%, Wales argues. This means, he says, that Neanderthals did not have to make tight-fitting clothes that completely covered them up.
We know that modern humans tended to hunt animals that would have helped them make thicker, snugger furs. The wolverine is a prime example. It would have made excellent trimming near the neck or at the edge of sleeves. Even today, wolverines are preferentially targeted by groups such as the Inuit.