I own an example of each, and here are my observations.
Reliability: Pretty evenly matched as far as the rifle itself goes. AK-style magazines are much more durable and impact-resistant, though; most mini-14 mags are made of thin sheet metal or rather brittle polymer, whereas most AK mags are built like dump trucks. It is HARD to find good mini-14 mags, and what works well in my mini-14 might not feed at all in yours, and vice versa. AK mags are of consistently good quality.
Accuracy: Pretty evenly matched, with good ammunition. A high-end AK lookalike (Vepr, Krebs) will probably best the average mini-14; a lower-end AK lookalike (Romanian, Chinese) will probably equal the mini.
Ergonomics: Out-of-the-box, I'd say the AK wins hands-down in stock design. But you can fit a mini with an aftermarket ergonomic stock to even things up. Both guns use the same rock-and-lock magazine system (equally non-ergonomic compared to an AR), and the mini wins hands-down on the ergonomics of the safety, which can be quickly and easily operated with the trigger finger. Unless you have fingers like Manute Bol or fit an aftermarket safety to the AK, you can't operate the safety lever with your hand on the trigger guard.
Ballistics: On paper, 7.62x39 wins at ranges inside 100 yards, and .223 wins past that. In reality, it's pretty even; the mini's inaccuracy keeps you from taking full advantage of the .223's flatter trajectory, and the AK's range-compensating rear sight makes the trajectory less of an issue out to 300 yards or so. If you can shoot well with an out-of-the-box mini-14 past 300 yards, you're either have a better mini than I do, or are a much better shot than me. The mini-14 has a much wider selection of bullet designs to choose from, though (from fragile 40-grain varmint bullets to low-drag 77-grain boattails, and everything in between). The latter quality gives the mini-14 the edge in a suburban home-defense role, since it's hard to find 7.62x39mm loaded with a fragile-enough bullet.
Aesthetics: Subjective, but I think the AK looks way better. Fit and finish is better on the mini than on most AK's, though.
Optics capability: Both are easy to scope (if you get the Ranch Rifle version of the mini-14, which I have). You can buy traditionally-styled scopes at Wal-Mart to fit the mini; AK POSP-style scopes are probably better quality for the money, but have to be mail-ordered. If you want a holographic sight, though, it's easier and much cheaper to get a Kobra for the AK than to fit a mini with a rail and buy an Eotech.
Cost of ownership: The mini costs between 50% and 100% more than an entry-level AK; the price is comparable to a higher-end AK. Ammo costs are now fairly similar.