It's cheaper. One 50 pound bag of thinset mortar will fill a 5 gallon bucket to the rim.
It will contribute to the overall strength of your floor.
It will set up faster (about 12-24 hours). Mastic will stay soft for days (maybe longer)assuming you are using a 1/4" thick mastic bed. With thinset you can feel safe that it's stiff however you cant tell with mastic.
There are two ways.
1. Use a paddle in a 1/2 drill (the easiest way). RPM's should not exceed 300 while mixing the mud. Fill a standard 5 gallon bucket with about 5 inches of water. Add the thinset slowly and mix. If you add the thinset first the drill will burn up from the resistance. Add water in small amounts as needed. Mix until fluffy. It should stick easy to a tile and hold it's shape (no sagging).
2. Use a margin trowel and mix by hand. The hardest way because it's hard to get the lumps out of the thinset. If you have a 3/8" drill with an adjustable speed control you can take your wife's cake mixing blender gizmo (I don't know what it is called) and insert one of the whips in your drill to mix the thinset with. You can only mix a little thinset at a time this way but it is a lot easier on the arms.
Don't allow the thinset to slake. Figure about 5-10 minutes for this to start happening.
I use my hands and weight to set tile but a mallot can be used. Just don't overdo it.
A helpful hint:
If you're using thinset, set your full tiles the 1st day then install your cuts the next day. Easier on the back and the nerves.