The biggest difference in Sullivans and Weavers is that Sullivans sets much faster. Many experienced fitters prefer it, because they can get their desired effect faster and with less product. I believe part of that preference comes from familiarity - Sullivans products have been around much longer than Weavers. On the other hand, Weavers dries slower and is more forgiving, so it may be a better choice for those with less experience. I know some of the best and most experienced fitters helped design and/or endorse the Weaver products, and they certainly get great results with them. But for most fitters that have been using Sullivans for years, there’s at least a learning curve for using Weavers, and many would just rather stick with what’s always worked for them.
Which particular product from either (or any other) line works best for me varies with a lot of factors - mostly the weather (temp and humidity) and the kind of hair I have to work with. I’d have to say if you told me I could only use one adhesive from now on I’d have to go with Sullivans Tail, but I know that’s sure not the right answer for everyone.