That's very different from the way the game was played from about 1980 onwards (perhaps a little earlier or later, depending on the locale and how thoroughly D&D had infiltrated the consciousness of the local kids). There was one Party per campaign world, and the same people would get together consistently to play. Obviously there were (and are) exceptions, but as a rule Party play became the rule. I think this had several impacts on the way the game was played.
First, it allows play to be suspended in situ, and picked up again exactly where it left off. The impact of this is enormous; no longer was it necessary for the cohors pro tempore to return to their base at the end of every game session after exploring the campaign's tentpole dungeon or some peripheral adventuring arena. It allowed for more wide-ranging play, where exploration was possible beyond "a reasonable distance" from a central city or town.
This also impacts entire sections of advice given to fledgling DMs in the Dungeon Master's Guide. When the party is always together (or reasonably so), sections such as "Time in the Campaign", which implores the DM to take good account of time passed for each character become rather superfluous. Naturally, if the players are all part of "the party", then A's fellows will wait around for him, or, better yet, accompany him and his elfin henchman to visit that oracle.
The reasons for the switch are manifold. Probably the most important practical consideration is that back when the game was first becoming widespread, it spread among individuals, who brought in their immediate circle of friends to play. Only the most hard-core gamers would cross-pollinate between groups (except at conventions and the like), so truly large groups of players where everybody wasn't free at the same time just didn't exist. Or, if they did exist, it was impractical for DMs of the day to run a game for a group of 20 players, even with a caller.
West Marches campaign springs to mind immediately, and of course Constantcon is an ongoing ad hoc cloud, for all intents and purposes. My own current Greyhawk campaign is consciously designed to use the cloud style of organization; we've got a number of regular players, but only a maximum of 6 can show up for any given game (that's a limit I imposed, as I find it the optimal number for myself to run a game for, as the GM). But I think that the shift from the cloud to the party is one that not only impacted the individual campaigns that utilized one approach or the other, but D&D as a whole, because it encouraged the style of play, and thus the design of adventures and even entire settings, that have come to dominate the hobby and industry today.
* Indeed, there wasn't even "the campaign" in the earliest days, but rather a number of different campaign worlds that players would bounce between willy-nilly, but that's somewhat beyond the scope of the point I'm making here.
** Which is not to say that there aren't plots in a cloud game, but they tend to unfold on a personal level, rather than a group level.