I've been wearing moccasins for maybe five or six years now, ever since I started running (and trying to live) barefoot. And the types I've used have all been from the company Minnetonka—the most readily available, in stores and now you can order them from their website.
Moccasins, or the models I wear, are 'zero-drop' at the heel, soft and comfortable, and the leather generally hides odor. I don't wear socks with them, not even in a Michigan Winter, where they generally kept my feet warm (I now live in Oregon and Arizona, so cold isn't an issue so much).
Minnetonka does offer a type of hardcore cold weather moccasin-boot, the Pug Boot, in the style of what Inuits and other northern tribes use. Leather on the outside and rabbit fur lining, with plastic soles. Super warm. I wore them with just my bare feet in Michigan blizzards and felt fine, even when the rest of me was freezing!
One thing I love about moccasins is how easily they slip on and off, more so than even any model huarache I have. Once they get stretched out and form to your feet, you can slip feet in and out without even having to reach down with you hands. So, for example, if I'm going to go hang out in a cafe and write and read, I may actually opt for my mocs versus my huaraches, simply because I know I'll be able to slip them off quickly and conveniently, and even a little discreetly too. And if I need to get up for something, so as not to freak out non-barefoot-friendly people, I can slip them on quick.
The biggest advantage of moccasins is that they are fashionable, yet still comfortable. A couple years ago, in Michigan, they even seemed to be somewhat 'cool'—at least with my young female students. And I've been told my female friends who are way more conscious about these things than I am that they look fine for a night out on the town, in jeans and a dressier shirt for guys. And, I think they offer a slight air of eccentricity. Perhaps. But the main thing for me is comfort.
People do use moccasins, and especially Minnetonkas, for other things. Some use them as slippers around the house, and in fact I did have a student ask once, in class, “Why are you wearing slippers?” And you can buy versions that have flannel or wool linings.
Some hunters like to wear them out in the woods, to be quieter. Which, you know, makes sense. They're what the natives of Turtle Island (at least in the northern climes) have been hunting in for hundreds, if not thousands, of years.
There are two kinds of Minnetonka moccasins I use and recommend. The first are the more commonly found in stores, and offer the double layer sole. These are the kind most often used as slippers. Note: there is a version with only one layer of sole, and though comfortable (like, feels almost barefoot) if wearing out on pavement in city, the bottom is going to wear through pretty quick. The double layer version may feel a little too cushy at first, but it'll mash down. Note too that these layers, these soles are of soft leather, and will get wet and soggy in any kind of wet weather.
For wet weather, I would recommend getting Minnetonkas that offer some kind of hard plastic sole. Women have more options here, as far as bottoms that are zero drop on the heel. For men, Minnetonka only offers one brand that is true zero drop, the Classic Moc. I did try one other kind for a couple years, the Double Bottom Hardsole, which look more like Docksiders, with a slightly raised heel, and they were ok, and even looked more 'normal' or acceptable, though one big problems is the inner lining was not soft leather, and therefore didn't absorb foot odor.
But the Classic Mocs do a good job of keeping the feet up off of wet pavement. The soles are a little high, all-around, and stiff, and if there are puddles, the leather can still get wet. Note: bought new, Minnetonka puts a leather heel insert inside that actually raises the heel a little. The inserts are just glued in, and I pulled mine out. This may have cause in bottoms to wear out quicker, and the inserts might just mash down after a short while, and off more shoe life. The next pair I buy I may keep them in for a while to see how they feel.
I did experiment with running in my moccasins. Not the dressier ones, but the soft-soled kind. And, they do work really well in cold weather for keeping the feets warm, even on snowy trails. They only problem was, again, they got wet, and the soft soles are kinda too soft, and wore out very quickly. On dry terrain they might last longer, though for that I'd just use some form of huarache sandal. But, if Minnetonka, or someone, could pair their basic double-layer moccasin with some kind of rubber or Vibram sole, that might be a nice cold weather running shoe.
Unfortunately, Minnetonka doesn't offer any moccasins with hard leather soles. That might be ideal both for wet pavement, and even for trail running. I have seen moccasins like this, one of my students in Jackson, Michigan wore some one time. I asked her about them, and she said that she bought them on her reservation nearby, but I didn't actually get the name brand, nor even which reservation. But they exist. [If you know what these are, please let me know down in the comments section!]
I'd recommend trying them on in-person at a store. Moccasins, being leather, tend to run smaller in sizes, but they will eventually stretch. Get them snug. Also: whenever I get a new pair, for some reason they chafe on my heels for a while. It's good, they're designed that way, to curve in and grip the heels. But fear not, just ease into wearing them and soon they'll be comfy. Like wearing slippers but out and about.