Does anyone have an idea whether there is a difference in meaning between 近几年 and 这几年？ I have the feeling that 近几年 is more formal - but correct me if I'm wrong!
As a native Chinese speaker, I would say that your feeling is correct. Usually, "近几年" is used in the news - like in newspapers and on the news on TV or in broadcasts. Anyway, they are interchangeable, so don't worry about the usage of them. They have the complete same meaning.
They are interchangeable and have the same meaning in almost all cases. However, 近幾年 is slightly more genteel and tends to appear in literatures.
In daily conversation, we generally say "這幾年".
近几年 = the several years near to (implicated:) this year or now
这几年 = these several years. These here often indicate the current year or now
近几年 - recent years 这几年 - these years
"recent years" refer to the immediate past. "these years" refer to the past, present and possibly has the implication that it may extend to the immediate future.