Leica M2 compared to a Nikon FM...
Bodies weigh almost exactly the same - about 600g for the FM, and 580g for the M2. The M2 is about the same size as the SLR body without the pentaprism or extended lens mount...
The Leica's top plate is a bit higher than the Nikon's, and it's a smidge fatter front-to-back, but that's about it apart from the obvious SLR protrusions.
In terms of weight in real-world use, I usually have a Tokina 25-50mm f4 on the Nikon...
...and a Voigtlander 35mm f2.5 Color Skopar (LTM with M adapter), and an M grip on the Leica...
Whether or not the Leica is considered big and heavy depends entirely on what it's being compared to. Certainly bigger and heavier than small rangefinders like the Olympus 35RC, but the latter isn't a full-fat interchangeable lens camera while the Leica is. I think comparing it to a compact SLR is more realistic. Given that I don't have a 35mm prime for the Nikon, and that the Tokina is probably bigger and heavier, I'd say there's very little in it. If the 50mm Pentax prime I have is more representative at 200g, the Nikon with a small prime lens would come down to just about the same as the Leica with the Skopar & grip. It would still be a bit more bulky overall, but in terms of actual shooting, it's neither here nor there - I shoot medium and large format as well, and class both of these as small hand cameras.
Aside from the functional SLR and rangefinder differences, the Leica feels much more refined. When I got the FM, I was surprised at how ratchety and noisy the film advance was - the Leica is nearly silent. The Leica's shutter noise is a very quiet snick, while there is the added mirror slap in the SLR. In terms of feel, the mirror movement is quite obvious - you can feel the weight moving around, but there's none of that with the Leica. I'm much more confident about hand-holding at slow speeds with the Leica than I am with the Nikon (eg, 1/4 sec).
With regard to the Leica 'thing', if it exists at all, it's probably mostly in people's heads. There is certainly the history - Leitz effectively invented 35mm photography as we know it, and it just so happened that the cameras they made for that format were of outstanding quality. The cameras have stood the test of time over nearly a century in a way that very few others have. In physical terms, perhaps the Leica thing is evident in the feel of the cameras - they're very smooth and slick. The FM feels like a well made, slightly workmanlike camera, while the M2 has class and refinement that's on another level. By 'class', I mean the approach to design and manufacture - the film advance doesn't
need to be that smooth and quiet, but they made it that way anyway. If a Zenit feels agricultural compared to the FM, then there is an element of the FM feeling agricultural compared to the M2.
While I quite like the history of Leitz/Leica, and admire the sheer quality of the classic products, it's still just a camera, albeit one that's a pleasure to use. For me, it's not really about the Leica 'thing', but simply about using a very high quality bit of kit. I started with a Barnack Leica - a IIIf - and was impressed with the build quality, but not so much with the quirky old-school controls and layout (fun to use, but not 'serious'). So, after some research, I went for an M2 because it was a 'modern' camera (lever wind, standard shutter speeds, better finder, etc). As a picture taking tool, it doesn't get in the way - extremely intuitive and transparent.
Ultimately, it's down to what you expect or want from a camera. In the 35mm rangefinder world, the Leica M is the top of the tree.