What is the cost difference between lifting (2') up the house and lower the basement? Which one is the better solution?

If you "lowered" your basement you would have to remove the concrete floor, dig down 2' and pour another floor. A big problem with doing that would be that you would be below the bottom of the footers, which would mean you would need to pour new footers while somehow supporting both the basement walls and the house. You are talking about a very major project.

Raising a house (or any other building) off its foundation is a very common project that many companies do. Basically, you just jack up the house, add 2' to your walls, and set the house back down. It is a large project, but not nearly as much work as trying to lower your basement.

I don't know what it costs, but raising the house would be a lot cheaper than lowering the basement.