Working styles aside, my main observation in living with Border Collies (though aside from a foster, never having lived with an Aussie) is that BCs are probably more "intense", but they're quieter about it. Aussies bark and bounce. BCs stare and stalk. Depending on your personality type, either can be extremely irritating. I don't find adult Border Collies very wild, they should be level headed and while they always retain some degree of reactivity (not in an aggressive way, but they respond strongly to stimuli from their environment) and excitability, I find them less "bouncy" and openly goofy than Aussies. Border Collies can be goofy, but usually as adults they are serious and rather stoic dogs with brief moments of puppylike behavior. I find that Aussies remain more playful and fun-loving throughout their lives, and Border Collies tend to take life too seriously if they aren't presented with a chance for some light-hearted play once in a while. While bcs like routine, too much of it can make them inflexible and obsessive.
Border collies are more prone to developing obsessive behaviors. Some just fetch the ball until they fall over, but some chase shadows, some watch TV, some dig... etc. Because they're so quietly intense, it's easy to ignore their obsessions or encourage them (because they keep an active dog busy) and you don't notice the behavior strengthening until it's a problem. I haven't noticed this so much in Aussies because they make their interests and desires very clearly known and it's easier for owners to nip things in the bud. Not to say that you can't prevent them with border collies too... just food for thought.
I wish I could give you a straight answer on training and obedience. Some people say Aussies are stubborn, some people say BCs are push-button, but I think it really depends on the dog and trainer. I've seen hard-headed border collies and extremely soft aussies. If I were to make a vast generalization I'd say that Aussies have more of a "what's in it for me?" attitude and Border Collies work because they're workaholics and enjoy doing things right. Border Collies can be completely crushed by failure, Aussies bounce back quickly from their mistakes.
I find BCs friendlier to strangers than Aussies, but less willing to work with them. My dogs are all very social, but they have no respect for anyone but me. I can't fully trust anyone with Eve (or trust Eve with a new handler!), because she listens only to me. When I'm gone, listening to other people is optional. It takes a month or so of living with her to earn her respect. Less for Dakota and Dash because they can be bribed with treats, but they are similar - definitely one-handler dogs. When I get another Border Collie, I will be putting a lot of work into teaching the dog to accept different handlers.
Border Collies are built lighter, smaller and longer than Aussies. Aussies usually have a thicker coat, and working line dogs of both breeds tend to be a bit smaller and lighter in bone and coat than their show-bred counterparts. There's a lot of variation of appearance in both breeds, and a big difference between the working and show lines - both in looks and temperament.
I love both breeds, they're both intelligent, loyal and responsive. I can see myself with a rescue Aussie someday.
Edit. Ooops. Posted before reading the rest of the thread. Tucker's link covers it and Sam's post on that thread is spot on in my opinion.
