I have owned both. For BP shooting there is no practical difference* IMO.
If you want conversion cylinders and the like, Uberti seems to have an edge.
* The main difference from my point of view is that Uberti makes some guns that Pietta doesn't, even when both companies list the same models in their catalogs. E.g. a Uberti 1862 is a small frame five shot .36 that is scaled to the caliber. The Pietta 1862 is a larger overbuilt .36 that really has none of the 1862 charm. Years ago I saw a repro 1862, wanted one, and ordered...and then had a big disappointment when I realized the Uberti and Pietta 1862 models were totally different. The Pietta .36 revolver worked fine and there is nothing wrong with it for shooting, it just isn't what I expected. I have since bought an Uberti version.