the main thing is Broadcast has the clipper that doesn’t do much good in streaming, unless you want to be excessively loud. (aka too loud imo)
You’ll want to use Live because -A- it’s capable of phase linear filtering, for higher quality sound, and -B- it can have any soundcard set for input and output so you can have an all digital setup. Also -C- it can load your encoder plugin directly into itself.
With "BAE" you would have to use an external encoder, and use the soundcard’s mixer to record from, and generally is asking for more chance of glitching, and lower quality sound, from that alone. BAE itself is made to be only for very low latency consumer use, which is why the sound quality for broadcast could be slightly better, and a great reason to go for Live alone.
Add in the speaker calibration stuff that runs AFTER any encoder plugins, and you get an optimized studio experience as well, or go for pure stability and don’t output to any soundcard at all, only the loaded encoders getting any processed audio. 8)