In its multi-player aspects, Star Wars: The Old Republic is less different from previous MMORPGs like World of Warcraft. Group size is smaller than elsewhere, only 4 people, but you'll still want a tank, a healer, and the rest doing damage. Flashpoints play very much like WoW dungeons, with the occasional dialogue being the only difference in a classic sequence of trash mobs and bosses. The bosses have special abilities, forcing players into a certain degree of "dance" to step out of the fire or avoid some whirlwind ability. The whole group gameplay is very recognizable.
Heroic quests are slightly different, being usually a lot shorter and having no or just one boss mob. Furthermore those heroic quests are located on the planets and integrate somewhat into adventuring on those planets. Flashpoints on the other hand start from a central location, the Republic or Imperial fleet. Some heroic quests can be done with just 2 players and their companions. And heroic quests appear all to be daily quests, while flashpoints can be run as often as you want.
One major difference between SWTOR and WoW for group content is the way groups are formed, and how they travel to the instance. In SWTOR you can flag yourself as looking for group, but there is no automated dungeon finder to assemble a group. Thus much of group finding happens in chat. There has been a lot of discussion about the dungeon finder removing the sense of place from World of Warcraft, but the other side of the coin is that getting into a group instance is a lot more difficult in SWTOR. If you are doing quests on a planet, you won't hear the chat in fleet where the flashpoint groups are formed. And if a group form via guild chat, getting from where you are to the instance entrance isn't all that obvious. Your emergency fleet pass only works once every 18 hours, so most of the time you need to "hearthstone" to the spaceport of the planet you are on, and then do a lot of running and waiting for loading screens to get from the spaceport via your hangar and your ship to the fleet and then the level where the instances start. And then you have to take the same way back in reverse to continue with your questing after the instance. Not very convenient, and thus there is a lot less group action in SWTOR than it could be.
The point where Star Wars: The Old Republic is much better than World of Warcraft with regard to multi-player PvE content is that there is less of a gap between single-player and multi-player content. In WoW some abilities are only ever useful in multi-player, for example a warrior's taunt. In SWTOR every player has a companion, and so can use "group" abilities and tactics even while soloing. If you play a tank, you *can* (and should) taunt mobs of your healer companion, and thus are better prepared for that situation if it happens in a group. Furthermore solo content in SWTOR is not totally trivial, as in WoW. Thus single-player in SWTOR prepares you better for group play, because you already learned how to play better, how to use things like interrupts, crowd control, or line-of-sight pulls from soloing. Of course that isn't 100%, but the gap is definitively narrower than it is in WoW.
While I haven't raided yet, I made a first experience with the raiding interface. In a regular group as a healer you see your health at a different location than the health of your party, which increases the likelihood of forgetting to heal yourself. You can fix that by turning on an option which uses the operations ("raid") frames for small groups.
That is if you can actually find a healer, or tank, for your group. Fleet chat is full of people looking for those for their groups. SWTOR does not have dual-spec, and thus the old problem arises that many players of classes which supposedly are tanks or healers rather use a damage dealing talent build during leveling, because that is more efficient.
As you can see a lot of the issues World of Warcraft had with grouping over the years are still there in Star Wars: The Old Republic. For me personally the biggest issue is that SWTOR hasn't moved beyond the idea of using dance-based raid content as major part of the endgame. For people who weren't into raiding in WoW, the SWTOR endgame doesn't appear to offer a solution. And for those who did raid in WoW, it isn't obvious why exactly they would prefer SWTOR raiding to WoW raiding once the "oh, new, shiny!" effect has worn off.