I also home brew and make wine. A big difference I have seen is most home brewers are ok with lots of headspace in a carboy/primary pail. You'll find that most winemakers are religious about topping off or racking down to smaller containers to remove head space.
Most wine is initially fermented in a pail w/ a loose fitted lid or towel covering it. Not under airlock. When fermenting with grape skins (red) or fruit, it is common to stir once or twice a day to push the fruit under. Wine appears to be less vulnerable to infections compared to beer.
Most home wine makers are not nearly as picky about yeast selection as what I have seen in the home brew community. Dry yeast is much more common than liquid yeast. And I think most of us just sprinkle it on the must to start fermentation (skipping the re-hydration step). Generally speaking white wines should ferment cool (say mid to higher 60's) and red wines fermented warm (mid/high 70s, low 80s).
Potassium metabisulfite (aka k-meta) is your friend with wines. It inhibits the yeasties, bacteria, and other nasty stuff since wine sits for months/years before bottling. In my experience many home brewers will not have heard of it or use it. Wine makers use it for both as a stabilizer in the wine (rule of thumb is 1/4 tsp per 5/6 gallons) and as a sanitizer solution (3 tablespoons per gallon) in place of star san. I buy it by the pound.
For what's worth I would recommend doing at least one wine kit. Did you start brewing from scratch w/ a complex all grain, stepped mash, and multiple hop additions recipe? If so maybe you can make decent wine from grapes right off the bat but there are a lot more steps to it, more points of potential failure, and it'll probably take longer. There are varying level of kits, the more expensive ones ($100+) will contain more juice/less concentrate, oak cubes/staves, grape skins, etc. and will yield a higher quality end product. White wine kits could be ready to drink in roughly 6 months, higher end reds usually take a year plus to become outstanding.