There does appear to be a difference, at least for me.
Radio Swiss Classic plays through the MC audio engine, and the Audio Path is populated with information. The "Start the radio" button doesn't switch to "Stop the radio" when I click it to start playback. I have to use the MC Stop button to stop playback. But the MC Play button doesn't start playback. I have to use the "Start the radio" button for that. Ctrl-Click works to initiate the popup.
Radio Swiss Jazz doesn't play through the MC audio engine, and the Audio Path is not populated with information. It plays directly to me default audio device. The "Start the radio" button switches to "Stop the radio" when I click it to start playback. The MC Stop button doesn't stop playback, while the "Stop the radio" button does stop playback. Ctrl-Click doesn't work to initiate the popup.
So some inconsistency in the implementation. Maybe at their end, maybe with MC. I can't tell.
Anyway, to get Radio Swiss Jazz playing, given the above issues, click on the headphone icon to the right of the highlighted track, which has a tooltip of "Hear Extract". Let that play, and when it finishes (if it does - it might just keep going playing the programme), click the "Start the radio" button. That worked for me, but as I said above, it isn't playing through the MC audio engine.
Radio Swiss Pop works the same as Radio Swiss Jazz.