Which Room has More Potential?

Sorry to hear about the wife.

without seeing either room, I would definitly go with the 13 x 24 room and lay down some good thick (cheap if need be) area rugs and padding underneath.

Proper speaker placement can overcome many room deficiencies, but an almost square room is pushing the limits.

And the bigger room certainly should have more potential to continue along with your improvements.

-IMO

Isnt it interesing how when the better half takes a hike all of the sudden audio possibilitys seem to expand?
;)

Now that your wife is gone you can close those 3 doorways with anything you want!

The 13 by 24 is the better size but the open doorways kill the acoustics. If you can't close off the doors with acoustic friendly doors. Go with the other room and tweak it.

set up your equipment in both and judge for yourself. nobody is going to be able to tell you which is better without seeing and testing the room for its oddities. spend time adjusting the setup in each room (speakers further from the wall, out wider, straight ahead or angled in, nearfield listening, etc...) to determine what sounds best to you.
have fun!

Assuming an 8-foot ceiling, the 13x24 has resonant modes at 70, 141 and 211 hz and a volume of 2500 cu ft. The 14x16 has resonant modes 70, 141, 211, 282, 320 and 353 hz and a volume of almost 1800 cu ft. It's my understanding that modes above 300 hz have little consequence, but I think the smaller volume may be disadvantageous. There is a free graphical mode calculator available online at RealTraps.com, and it provides recommended room dimension ratios near your own. For instance, if you could construct a 5-ft deep antechamber, you would have a nice set of dimensions. Download and check it out.

Thanks for the comments everyone.