Let's Tries are just demos. It's why it's called "Try," because I'm not playing a full game. I also figure there isn't any reason for someone to ask me to play a demo that isn't theirs; most people only ask me to play their demos for feedback to help them in completing their game. A Let's Play though is a complete game. Since the game is done, I can give my opinion, but it probably won't affect the game since it's already completed (unless the developer is still actively patching it). So most of the enjoyment comes from my crazy commentary. A Let's Play can be requested by either the developer or someone else who may just want to see my reactions or hear my thoughts.
Anyway, I guess you could request a demo (ie Let's Try) of someone else's game, but I think it would be rather odd. ^ ^ ;;