No. There is no online story mode. Your character interacts with several SP story characters in events that 'occur' before the SP story, but have NOTHING to do with the story in any way. Its basically just an open world, do whatever the hell you want, online experience.
Rockstar has added several vehicles every month or so, as well as adding bank heists (which require 4 people and have setup missions leading to the actual heists), they've added CEO's (where you and up to 3 'associates' steal or purchases various cargo and have to transport back to a warehouse, while becoming an active target to every other player to try to attack. Once your warehouse is full, you can go on "sale" missions to make lots of money. The CEO keeps 85% of profit with the other 15% being split amongst the associates.), and most recently they've added Motorcycle Clubs (where you and up to 7 other friends, run various illegal operations such as Weed Farms, Cocain and Meth Labs, and Money and Counterfeit ID operations. You need to keep these operations stocked with supplies, and sell the goods once their inventory is full. Every time you have goods, be it supplies or inventory, you are a target for other players in the room.
CEO's and MC's occur in the open world. All the servers are maintained by Rockstar. There are no "user created" lobbies, and although you can join an online Rockstar server by yourself, with friends only, with your crew only, or jump into an open online server, all of the CEO or MC missions are only available in open world sessions, where there are players who can, potentially, foil you. Sometimes they do, sometimes they don't care.
Heists start a mission like the original GTA Online missions. They're not open world. They create their own lobby, just like races, death matches, and other PvP missions.
I've never paid too much attention to the 'pure' PvP side to GTA, as it is not my thing, but I know that they have added things like Capture (essentially capture the flag, but with briefcases), and they've added Stunt Races, which are actually a ton of fun. Unrealistic with massive ramps, jumps, loops, tunnels, windmill style opsticles, etc, but much easier to navigate then the narrow street courses that GTA originally gave you.