I have two bikes, a hybrid commuter with skinny tyres, full mud guards & rim brakes, then I have a 29er hardtail MTB with hydraulic disc brakes and off road tyres. Which would be best, for commuting (with a minimum modification cost) in the snow? (this is likeley to be a small-medium amount of snow, perhaps having been compressed by other vehicles/walkers)
Having lived in a few parts of the US, I've found what people call small to medium to be drastically different based on the region ("oh yeah, we just had a light snow last night", and i look outside to see 4+ inches versus "it snowed like crazy last night" and i look outside to see a half inch and the whole city has come to a grinding halt), so its a bit hard to judge how much snow you're going to encounter, so I'm going to assume 2-3 inches fairly regularly (if it snows once or twice in the winter where you live, who cares?).
The ideal winter bike should be cheap and preferably somewhat disposable. Full fenders are good since they'll protect you from slush and what not and your drivetrain. The slush and snow and grit will still get into your drivetrain and what not even with full fenders, so you do have to make a habit of cleaning the bike off regularly in the winter (which can be tricky depending on where you live), or component life can be drastically shortened.
If you expect to encounter ice, studded tires are advisable (Peter White's page is an OK overview for this: http://www.peterwhitecycles.com/studdedtires.asp , though Nashbar does (used to?) sell very cheap studded tires made by Kenda - not good for off road, but perfect for commuting). I'd say have these on hand if you can - in my experience, snow often leads to ice. Just don't try riding like a maniac or you're wear out the studs. If you can't afford them, get a cheap set of knobbies and run them at low pressure and be careful (always be careful when theres snow+ice around). Be careful with sizing since studded tires of the same "size" may need slightly different clearance.
As for the hybrid, it depends on what kind of hybrid it is - if it isn't twitchy and can use "large" tires (say, like a Trek 7.2 fx or a modern Specialized Sirrus), it should be OK. If not, I'd use the mountain bike and try to fit some fenders on it (probably you'll be only to fit things like the SKS X-tra dry, X-board or something similar, but its better than nothing) and take extra care to clean it up regularly. Discs would be nice for winter commuting at times if your rims are covered in slush, but a good V or canti setup is fine as well. Overall, if you have to use one of your two current bikes and both fit the conditions above, use the cheaper one or the one you're more comfortable riding in messy situations.
The best option IMO is to find a cheap old rigid (one less thing to go wrong - some of the cheaper suspension bikes make horrible noises in the winter and its just not necessary) mountain bike (think 90s specialized hardrock, giant boulder (I've use a 2003-ish one of these as my commuter for the past few years, since its fairly comfortable, pretty cheap to maintain (though things rarely fail, even with cheap (acera-x level) components), fits full fenders and a rack, fits reasonable sized tires and nobody wants to steal it!) or whatever - the spec, weight is far less important for a beater than for your primary bike, just a decent fit) probably at Goodwill or something (these are quite cheap to find about now - if you live near a college campus and go to a bike shop which sells more cheaper / used bikes, you'll be able to select some from there at a bit of a higher but reasonable price), give it some elbow grease (probably a new chain and lube and new brake pads, some various adjustments), fit it with full fenders, some lights (extremely important in winter) and some knobby/studded tires. Replace components as they start being not good. The total cost for this could be less than 100 dollars and two - three hours of work if you're smart about it and prone to getting distracted (so you can then invest in some winter clothing!).
I do not know what you mean by "on budget". Commuting 25+km one way throughout the year, I stopped experimenting with e.g. low tire pressure on otherwise unmodified bikes, or using cheap bikes during winter. Considering the (economic & health) risk you take with minimum modified, on budget bikes, I recommend to switch to public (or other means of) transportation to stay on budget.
For winter biking, IMO cost savings should not be highest priority. Otherwise you pay the price when you get hurt, hurt others, or end up with damaged equipment.
The only way for me to be / feel as secure during winter as without snow and ice are studded tires. If your winter conditions are as light as I understand, the rim brakes on your commuter bike should be OK. However, especially when the daily temperature variation allow ice to melt and freeze again while your bike is parked, those brakes can become hard to use. And I would spend some money on lights, which will be battery lights if you are out for cheap.