There seem to be some people who mention the newest software updates with the MC11 adapter can allow some Sony cameras to operate without restrictions that typically are forced when using adapters - but I can't say for sure - some folks I know who used the MC11 say the same restrictions exist. Namely, you are restricted to 3 frames-per-second burst mode if you want to maintain continuous focus, and you don't have access to some of the focus area selections such as expanded flex spot or zone.
Otherwise, both adapted lenses should perform roughly equally with regards to single-focus speed and accuracy, and continuous focus ability. Both should be usable for still birds, and a bit challenged for birds in flight, though still capable if you can live with 3fps.
Both will struggle mainly with big focus transitions - meaning if you focus on something at minimum focus distance, then try to quickly transition to a subject at infinity focus, or vice versa - you can suffer a delay of several seconds while the camera's focus system tries to make something of the complete blur on the screen until the lens starts to get closer to the subject and it realizes there's something to focus on - then it will snap-to quickly. This is an issue with almost all adapted lenses on mirrorless bodies. A workaround most will use is to 'pre-focus' a bit when walking around - if you just took a closeup of a spider on a web from 8 feet away, and are now about to walk out into a wide open field to look for birds 100 feet away, find a tree or bush in the distance to focus on first, that way the lens is pre-focused closer to the distance you will likely be shooting. When you do see a bird, it will focus much more quickly.
The Canon version has one advantage over the Sony A-mount version - it has in-lens stabilization. That's one thing to consider, especially if you will be shooting handheld in poor light and may need slower shutter speeds.
I've used the G1 version of the Tamron in a Sony A-mount on my A6300, and it does just fine - I shoot in pretty good light most of the time and have a pretty steady stance, so stabilization hasn't been an issue for me...I've shot some birds in flight, and some low light situations in dark forest cover where i needed ISO 6,400 to keep the shutter speed at 1/600 to 1/1000, and the results were still pretty solid. I've got the FE100-400mm GM lens for the camera too, so that's really my primary wildlife and birding lens for this rig - the Tamron 150-600mm was already in my collection from my A580, so I occasionally pull it out for fun.