I went to both in August.
The food at the Leaky Cauldron is supposed to be traditional British — I thought it was very poor quality, to be honest. It’s like British school food: dry, tasteless pies cooked and served in thin aluminium tins. The food at the Three Broomsticks is American — ribs, chicken, corn on the cob. Three Broomsticks wins on food, but neither’s great.
Atmosphere-wise, I preferred the Leaky Cauldron — I’m not sure I could articulate why.
You can buy bottles of Gillywater for the hilarious price of $4.50 — it’s literally just water.
Anything with Butterbeer in the name is worth trying, however. I enjoyed all the various forms of it very much.
(Don’t know why they’ve come out sideways.)
If I had to choose I’d go for Leaky Cauldron — it just felt more authentic for me. And you’re not really going for the food.