Right out of the box the difference in user experience will be substantial. The base model MacBook Air has a very fast SSD (solid state drive) and 8GB of memory. The base Mini has a slow mechanical hard drive and only 4GB of memory. So the base MacBook Air has a disk that's about 6 times faster than the base Mini and it has twice as much memory. This makes a big difference.
I have a base Mini and it is a really sluggish feeling computer. For example, it takes about 15 seconds just to open the System Preferences. This happens almost instantaneously on the MacBook Air - I have owned two MacBook Airs and they literally run circles around the base model Mini. I would not recommend the base model Mini for an all around computer, most people will find it very frustrating. I use mine as an iTunes server and it works fine for that (it just runs iTunes with home sharing for my two Apple TV's and Macs).