No, it will not be much a difference from your Badger. The Paasche H is a real classic and my first AB dated back to the 80's. It is a external mix, single action AB. Both are good beginner AB for occasional uses.

First, consider what you are NOT happy with the Badger. Then, you can plan the upgrade. If you airbrush a lot, invest in an internal mix brush.

After many years with the H, I finally change to an Iwata HP-CS. I never want to look back. Yes, it costs 3 times more than the H. The smooth and precise control, the ease of cleaning are worth every penny.

I would probably second to a large degree keilau's post.  I own a Paasche H and am extremely pleased with it.  Having said that, I don't see much difference between it and the Badger 350.  Although own a different model Badger than the 350, I just can't believe that you would see a huge performance return on buying the Paasche.

As to what you might not be totally pleased with the Badger, if it is lack of fine line control, you could see about getting a fine tip (assuming that you have only a medium tip).  I know with my Paasche, going from the medium tip to the fine makes a world of difference in paint control.

If you simply want to upgrade, I do think it would make more sense to get something that is significantly enhanced.  Iwata comes highly recommended.  Badger 100 is not a lot more money than the Paasche H, and also is highly recommended.