I don't think it's that different, unless you're doing something with the raw ingredients (crushing grapes for example vs a grain mash). Are you looking at a kit, or raw ingredients?
Kits tend to be designed for 6 gallon containers, so if your carboy and bucket can hold that, you'd be fine.
If you're using grapes, you may need some different equipment. Including the possibility of campden tablets.
Otherwise, primary bucket, secondary carboy, racking cane & hose. Same as with beer.
I haven't had a problem with wine in beer bottles, other than if you're a stickler for presentation. There could be an acid reaction with the lining of the cap in the long term, but you'll have to find someone who tried to keep it that long in one. I used beer bottles because I liked the smaller serving size. I didn't have to bother with oxidation when I only wanted one glass.