Difference between Media and Asset module
Looking at this issue in the media module issue queue, the main technical difference (apart from UI) would be that the Asset module creates its own entity while Media uses the core managed_files API. In other words: Asset is 'an island' and other modules must specifically support Asset module. Integration with media module is easier because it uses core infrastructure.

There actually is a request to merge parts of Asset &amp; Media code.

I took the module for a quick spin, it looks pretty easy but less flexible then the media module. However, I always found the media module configuration very complicated (file styles) because of its flexibility probably.