If you're looking for price as a major factor, I would suggest the audio interface all the way; mixers get far more pricey than audio interfaces as you go up in quality and an interface will give you all you need for home recording as long as you have software on your computer to mix with.
Mixers tend to be more practical live though, as the speed with which you can mix instruments will be a very valuable live asset.
If you plan on producing records or doing lots of live performances in the future, I would suggest getting a mixer earlier on; the audio difference between them and an interface is very small, even indistinguishable to most.
If you're looking for a more budget-friendly recording unit, the interface will give you what you need with more ease and less price than a mixer.
If it's within your capabilities, I would try to use both in person and see what you prefer.