Not entirely sure what you intended here. It is obviously true that their original game was not conceived with live streaming in mind. OTOH, is the cast somehow unaware that they are now doing a live stream? I think not. I also doubt that [MENTION=6801558]robus[/MENTION] was confused about either of these.
Yes, clearly different - the question is to what the various differences are attributable. The problem that I have with many comments made about CR is the frequent certainty expressed about the performative (and other) aspects of their game being solely or mainly due to the fact that they are live streaming it, as opposed to them being a bunch of actors entertaining themselves while becoming more familiar with the game and more engaged in a very regular playing schedule.
I think in order to judge this reasonably one really has to have watched most or all of the series, at least Campaign 1, to see the evolution - their game certainly did not take its current form in the first streamed episode. Also, there are additional sources that may (or, depending on one's evaluation, may not) be revelatory of the cast's attitude and approach. These include Talks Machina and the panels that the cast does at cons. And if one wants to dig further, there are Between the Sheets, the Fireside Chats, the Critmas videos (including one featuring Matt and Marisha shuffling around at home in pajamas), and a few soliloquies by Matt, Taliesin, and maybe Liam, also. It could be claimed that all those are simply extensions of their performance, and I can't disprove that, but it does not seem like the simplest or most likely description.
Anyway, there are clearly things that have to do with the stream - production values have become much better and they wouldn't be doing general announcements or sponsor bits in a home game. The central question of the extent to which having an external audience causes them to play differently than they would in a home game with similar experience and a similar schedule is IMO a bit murkier. The fact that they are actors is often cited as an argument that it is inevitable that they would play to the audience, and it seems likely that there is some of that effect. However, as actors I would think that they would also have a greater facility at putting aside the existence of the audience if they wished to, and my fairly strong impression, on the basis of having seen a fair number of the sources mentioned above, is that they do, in some sense, want to.
More precisely, it seems to me that they conceive of the audience in a different way than one would the audience for a play, or a comedy routine, or some other kind of produced show. It's much more personal, sort of as if they had invited a (now rather large) number of acquaintances into their living room to enjoy whatever shenanigans happened to transpire. The cast is just as likely to fawn over fans as the other way around (at least until it exhausts them), and on the other side of the coin they react with some indignation to critiques of their game from fans. They (Matt usually more diplomatically, and Liam less so) have made it clear on a number of occasions that they have zero interest in altering their game to suit audience tastes. In short, there's certainly the occasional flourish directed at the stream audience, but only as it strikes their fancy, not as an obligation.
Indeed, and if polished-performance-for-audience were really at the top of their priority list, there would be a lot less cross-talk and fewer one-on-one conversations in parallel with Matt's narration or interaction with another player.