Few things
By now plenty responders commented about buying cash vs w/ a mortgage. IMO there is no right or wrong, but what is right for you. ex, If you are someone who is looking to retire soon and need to put emphasis on cash flow (CF) it favors one strategy over the other.
BTW, between all cash or a 20% down there is a VERY nice middle ground - 50% down - which gives you both a better CF while still leveraging - just an idea.
As for your questions which seems to be mainly Austin vs. Nashville.; I'll assume you mean the metros of each and not the actual city.
I know both metros VERY well.
Few pointers:
Both metros are about the same size population-wise.
Austin (TX) property taxes are approx. 2.5 higher than Nashville
Ins. rates are approx double in Austin vs. Nash.
Austin had appreciated in a much higher rate than Nash. in recent years.
100k purchase price in both would be a crappy property.
For nice good schools, middle-class area you are looking at 175k +/- in metro Nash and 250k+/- in metro Austin.
All in all for many reasons Nash. wins, IMO.
That doesn't make Austin bad, not at all. There are many good things working in its favor.
BTW, as someone who has been investing in all TX main metros sine 2004 I'd say if in TX . . . than Dallas and not Austin.
BTW, what made you decide Austin and Nashville to begin with?
Just to be clear I live in CA not in Nash or Aus or DFW just in case it sounds I'm biased.