I think in terms of who's missing out on what, there's always the matter of games that are exclusive to a specific platform, so I'd say PC gamers are missing out on God of War, Halo and Red Dead Redemption for example, same way console gamers miss out on a bunch of exclusives from competing brands (Microsoft vs Sony vs Nintendo).
In terms of game mechanics, I don't think there are many differences. Maybe in the VR front, but then again one could argue it's more of an external thing to games. I think the main difference is still in the kind of services and ease of access the different platforms provide.
The thing about PCs is that games are muuuuuuuuch cheaper in platforms such as GOG and Steam, mainly because they have so many sales and reduced prices after some time of having a game released (not to mention way more giveaways with sites like humble bundle and even GOG every once in a while), which doesn't happen often in digital marketplaces on consoles (well, at least in Brazil, don't know if that changes depending on the country). Also, on the PC one can play games from several different "generations" without having to keep a bunch of consoles or waiting for backwards compatibility (meaning I could switch from Jedi Knight Dark Forces to The Witcher 3 in a minute), so I guess console players miss out on that. Then there's the whole thing with graphics, having to pay for multiplayer or not, etc...
On the other hand, console games are as easy as putting a disc on the tray and playing it. At least that's how it used to be, but games got so big now you need to install them just the same as with a PC, however, the difference is that it is guaranteed to work on the console, whereas you might run into issues on the PC sometimes.
I might have gone on a tangent here, but that's how I see the difference between platforms.
TL;DR: the main difference, in my opinion, lies in the kind of services provided and exclusive games.