Do native speakers of different languages make different mistakes in English?

This is a much more complex and fraught question than it may seem. There are really two questions:

1. Do speakers of different languages make some errors influenced by their first language? The answer here is, yes, of course. Sometimes it's called L1 interference or transfer and has been studied extensively. Contrastive linguistics (mentioned by @jlawler) thought (erroneously) that this could even be used as the basis for all language teaching. The types of errors influenced by L1 are most obvious in pronunciation but also pop up frequently in idioms, vocabulary choice and also in much of syntax. Morphology is much less susceptible to L1 interference in non-related languages. So an English person learning Russian will make similar mistakes to a Spanish person learning Russian. But a Czech person learning Russian will be more influenced by their own morphology which functions on similar principles and sounds similar.

But you can also ask:

2. Does your native language determine absolutely the type of errors you will make in English? Then the answer is no. Not all features of native languages transfer over. And some features you would expect do not. So for example, even native speakers with rich verbal morphologies continue to make mistakes such as 'Jane walk.' even at intermediate level. So even though English verbal morphology is almost non-existent, it is still a challenge for them. Also, speakers of languages with very different syntax from English don't often transfer their word order into English in obvious ways. The interference may occur only in certain contexts and with certain constructions.

If you want more details, I recommend Rod Ellis's classic 'The Study of Second Language Acquisition'.