First things first, one huge thing I like about Goldeneye is the controls. I can honestly say it has the best controls for a console FPS (that I've played anyway, maybe there's better). Of course, that's with 1.2 Solitaire, the other control schemes are pretty bad, but at least it gives you the choice, a nice little thing that many modern console games don't do. All the buttons you need are right next to each other, while also being spaced out enough to not press multiple at the same time, there's no need to stretch your fingers too far for anything, and it's good to use for both left-handed and right-handed people, thanks to movement being mapped to both the d-pad and c-buttons (in 1.2). The camera is also very smooth and responsive, something not apparent in most console first person games - the analog stick is not as responsive as, say, a mouse.
This is most likely all due to being more of a fast-paced game compared to modern fps games, which tend to be "get shot a couple times, run behind cover for a few seconds to regen, rinse an repeat". No strategy is involved and forces the player along a linear pathway, rather than letting them explore and devolving gameplay to nothing more than 'point and click'. Exploration and fast paced gameplay are a couple of the many reasons I love 90's shooters.
Anyway, the gameplay of Goldeneye is simple, but refined, as well as having interesting features which work well and compliment each other. The player can take two different approaches - balls to the wall, full guns blazing; or being more stealthy and tactical. Both are just as effective as each other and both provide the player with fun ways to play in such a way. Want to make a bunch of noise but use powerful weapons? Dual DD44 Dostoveis are fun as hell, or even an RCP-90. Your trusty silenced PP7, D5K Deutsche or even fists are good for a stealth approach.
Weapons as a whole are excellent in Goldeneye as you're given a fairly large variety and the majority are fun to use and have fantastic sounds, hell, even the bog standard KF7 is great to use. The shotgun, Magnum and Moonraker Laser are particular favourites. The only real complaint in regards to weapons is that even though you have a variety, this variety doesn't present itself often enough - most levels will have you using the same few weapons and a lot of the really cool ones like the Magnum and automatic shotgun are pretty rare (though it's understandabl for the OP weapons like the Golden Gun or RCP-90). It would also have been cool to use the laser watch outside of the Train level, there were several moments where it could be used.
Level design is exquisite - maps are open-ended and give you several different ways to complete objectives, allowing you to play in different styles with few restrictions, as mentioned before.The level designers had never worked on a game before, which made them work in this more open-ended style, a trait which greatly benefited the game. Compared to a modern James Bond game like Quantum of Solace (which I quite enjoyed to be fair), it really gives it a more rich, unique experience rather than being a generic corridor shooter. You get to be familiar with the levels in Goldeneye and you get to learn effective routes through it, whereas with something like QoS, the levels are more forgetable and you walk past everything and never see them again, making most places appear fairly dull (at least in comparison to Goldeneye). Take Facility for example, you get a choice as soon as you start the level, just before you drop from the vent you can decide whether to take out the guard in the stall stealthily - which requires precision - or just leave him and most likely alert him later. You're then given a few routes to take and as you play, you can experiment to see which is most effective. Something as simple as that encourages replayability - and I've been replaying Goldeneye since the early 2000's.
Every level also feels different and unique (though they use Surface and Bunker twice) and follows the film closely without feeling overly forced like a lot of liscensed games. The fact that they allow for multiple approaches to a situation, as said before, is a huge plus and makes it great fun trying to find the best ways to beat the par times. Most levels also have a sense of simplicity to them while still feeling fleshed out and not feeling boring. Archives is a good example, overall it's a fairly short and easy level, however the space it occupies is utilised extremely well. Every room has a purpose, though not all are relevant to the game - the design is realistic without being obnoxious like so many "realistic" games tend to do. Goldeneye's levels remain fun to explore, especially for secret stuff like the movable walls in Archives.
And as for music, does much even need to be said? Let's just say there's a reason Grant Kirkhope is one of my favourite game composers - https://www.youtube.com/watch?v=oRhRCvEeYvk. It's hard to explain, but each song really fit's its level while also really having a "James Bond" sound to it. Take, for instance, Bunker - https://www.youtube.com/watch?v=xxW7jRDwmuY. It gives the sense of being stealthy as well as bringing thoughts to technology, something which may be a more subjective view. Or Depot - https://www.youtube.com/watch?v=b0PZqy4ewk8 , which has a very "military" sound to it.
That's all the good things done, now for the huge flaw Goldeneye has - the AI is absolutely piss poor. Goldeneye feels easier than what it really should be, even on 00 Agent difficulty, they're quite bad at times. My favourite example is in Archives, where you can literally stare a guard in the face - right after shooting up the previous room, mind you - and he won't notice you because a railing is in the way. This is made stranger by the fact that there are similar situations in Caverns, yet those guards actually spot you. Overall, it's not too big of a problem, the fact that every other aspect of the game is near flawless more than makes up for it.
In short, Goldeneye's pretty fun.