Sure... the difference is in the bluetooth profile that is used. The app that you're talking about is relying on the handsfree profile, which is mainly used for telephone conversations over bluetooth--it is monaural and very low quality.
When you select "Set as wireless stereo", you are using the A2DP profile which was designed for music and other high quality audio sources.
The two profiles have completely different intended uses, but both can be used to listen to music (assuming your bluetooth listening device also has both profiles).
The bad news is that unless your stereo headphones were designed to be able to activate A2DP using a button (some models can, some cannot--it's all in the way the manufacturer designed it) then your only option (that I know of) is to use the "Set as Wireless Stereo" menu command.
I know it's stupid... hopefully Microsoft will fix it eventually.
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