I just left Three for BT Mobile. Haven't received my sim yet so cannot comment on the actual service, but here is my reasoning for going with BT Mobile rather than EE directly:
Data Allowance/Pricing: EE only offer sim-only plans of up to 10GB of data. As of right now, a 10GB EE sim is £26.09pm, a 20GB BT sim is £25pm (or £20 if you are a BT phone/broadband customer).
Data Speed: BT Mobile is capped at 30mbps down/15mbps up. While I'd rather it wasn't, I'm coming from Three where my average data speeds on 4G are much lower than this anyway, so I'll still be getting a faster service even with the speed limitation. Also I don't think there's anything I can't do with 30/15 on my mobile so it's not so bad, it's more the geek inside me that wants to run speedtests that cares
Tethering: Being with Three on a legacy AYCE contract, I couldn't tether in the first place. To me it's not a big deal, the only time I like to tether is when my home broadband goes down and for those rare times I've actually used one of the EE 200GB sims seen as they always seem to be giving them away. I understand that tethering is a deal breaker for a lot of people, but I'd much rather have 20GB handset data than 10GB of tetherable data.
VoLTE/WC: As it stands, BT Mobile along with any MVNO as far as I'm aware, cannot use VoLTE or WiFi Calling. This will probably include the 800MHz band. This isn't a handset limitation anymore as with carrier updates on iPhones and the same sort of thing now on Android Marshmallow, any carrier should be able to do VoLTE/WC as long as the host network allows it. What with BT buying EE, I'd say if any MVNO is going to get these features first, it's going to be BT Mobile.
To be honest, I'd rather not have any limitations and join EE, but with a maximum sim-only plan of 10GB it's not happening, it's simply not enough data for me. I already own an iPhone 6S Plus and I'm not about to take out a handset contract just for the data allowance.