This is faster :) No Apple Push to get in the way
Our push notifications get delivered the instant it hits our relay. I doubt it. It could be slightly, but likely not noticeable.
At least 500ms slower due to the push system pings. But much more if the phone is locked. Push is VERY slow. Especially if the device is sleeping. You also don't see reactions from other relays, especially DM relays that require auth.