Since this is the hunting forum, 5.56 (223) is a fantastic coyote/varmint round, and the only one I use for that purpose right now. Flat shooting and cheap like borscht.
The 7.62x39 undoubtedly works well on deer sized game. I'm sure it would be fine for bear and moose at modest ranges, the biggest limiting factor I see is a lack of choices in bullet weight for hunting ammunition. A good 150 gr .311 bullet would be nice. But I have numerous other hunting rifles, all more accurate and harder hitting than any SKS I own, so it remains a cheap plinker in my collection.
I think the 7.62x39 would really shine in a decent bolt action with handloads, something like the CZ or Zastava.