As a performer ive found that its less about an intrinsic difference and more about how you think of your audience, e.g. the difference is with me rather than with the audience and what my goals are.
In the specific I've actually been spending a lot of time comparing my current group dynamics with ones I've had in the past, and it made me realize that part of our problem is that we don't play as if we value one another as an audience (on the player side anyway, GMing can have a specific culture) essentially everyone is playing for their own reasons, and we often aren't fans of each other's characters so in turn we don't get the character interaction we want or need as everyone focuses on their character's interactions with the GM and the NPCs and neglect to support it when someone else is 'doing a thing.'
Since realizing this I've became a player for 8 months or so, and sure enough, playing my character to interact with things other people are doing, being fans of their characters, talking to them within the party, making sure we have conflict (while making it clear its between the two characters, and ensuring the character isn't disrupting things we want to do) and generally treating the other people around the table like an audience I mean to entertain, impress, and engross like in my other performances (and of course GMing) is working wonders.
More to the point of the thread, I think that's the secret with Critical Role, you can see that they're very much fans of each other's characters, so they interact, they push to get information, they have heart-to-hearts, they support it when one of them is trying to do a thing with their character, they look for ways to link the character's together or establish interparty banter, they all treat the other players as part of their audience, and they treat themselves as an admiring audience for the other players. I think its something the Critical Role cast has experience doing and thinking because they all take mutual pride in their performances, but its not something you have to be a professional with a non-participatory audience to do, you just need to take the participating audience seriously as an audience, and they need to take you seriously as a performer.