I've only been to Tbilisi and Yerevan, both have a lot to offer but for me Tbilisi beats Yerevan all day long.
Tbilisi is well connected by buses, underground metro and shuttles marshrutki, and there is funiculer which was working when I was there. The whole city is like a mix of old beautiful ruins and modern stuff, it's got this unique vibe about it. There are also water spings everywhere in the city streaming from the caves and people drink from the them without any worries. What to see: Old Tbilisi, Botanical Garden, Museum of Art. What to eat: every Georgian thing on the menu and you will be blown away especially meat skewers, grilled veg, traditional cheese and wine. I still remember cheese pastry khachapuri and I would go back just to have it again. You will also see traditional bakeries where Georgians buy bread, massive airy discs, oh my God, they are so good.
I would strongly recommend getting a car ride if you will be going to Yerevan from Tbilisi and vica versa. I was with 1 other person, we paid USD50 one way, whizzed through mountains, had an amazing lunch, and arrived in Yerevan in like 4 hours.
Everywhere we went the locals were super nice.
Whatever you end up doing, the Caucasus is great, so embrace it and enjoy.