From what I've seen it's more of the same. It's not like the old days.

Yes, they are, at least in the sense that it's (largely) the same code being compiled into different binary types for different operating systems - in a number of cases, it's even the same (Windows) binary running through translation (Wine, and various derivatives of that, such as eON) on OS X and Linux. Back then, they (usually) had to rewrite the game for each system.

There are some differences though. Since the game on Windows is likely using Direct3D for its 3D rendering, those instructions have to be replaced with OpenGL instructions for Linux and OS X (or Mantle for the latter), and since the windowing systems are different, there are differences in how the game creates its window (whether "windowed mode", "full screen", or "borderless window", it's all a kind of window). Also audio, input, and networking. But everything inside the game itself (ie. not communicating with the surrounding system) is very much the same on all platforms, even when extending to consoles.

Of course, with all the ready-made multi-platform game engines around that mostly takes care of these differences, the game developer don't have to worry much about them.

Thank you. I'm thinking of trying Linux again some time.

They generally use the same source code and just have to write OS specific and/or arch specific differences and fix bugs in the code that surface on the new platform that may have not been a bug on the original platform or just didn't show up, but essentially they should be the same game overall. It's slightly different going between consoles and Windows/Mac/Linux often due to a variety of reasons.

One of the reasons games were so greatly different between C64/Amiga/DOS etc. back in the day is that the video hardware, audio hardware, input hardware, the microprocessor and just about every other aspect of the system was completely different so little to none of the original game code would work as-is on one of the other platforms. They essentially had to rewrite the games from scratch to the capabilities of each computer. Nowadays games are usually written in high level languages like C, C++, or even Java/C#/Python/Flash and the languages themselves and the interfaces the developers choose to use are often very portable if they are smart, so it's more a matter of adapting existing code to the new platform than rewriting the entire game from scratch as it was in the old days.

Keep in mind too however, that the game development company that created a game is quite often not the same company that ends up porting the game to Mac or Linux, and the quality of the results can suffer immensely even if it visually appears to be the same game.

Every Linux game I've played so far is exactly the same on Windows. Amiga and C64 were very different in graphics and sound, this is not the case though :-)

Instead of DirectX they use OpenGL which offers the same features as the former. If you're a complete beginner I suggest Linux Mint, forget Ubuntu and all the nonsense it mentions and if you're mildly experienced then Debian is a solid choice you can't go wrong with.

if it helps you, the gameplay feels the same. It's not like that the graphics, menus, music or sounds feels in any way different then back in the day. That's the good news. Bad news is maybe it's that often it's the same, which means games are just quickly ported not well optimized (example here is the mac pots of metro, which have massive performance problems with nvidia graphics cards).