I would say that "old" refers only to age, whereas "antique", by its use, refers to something that is of value or interest partly or totally due to its age.
Something that is "antiquated", however, means something that is no longer used or no longer useful, generally due to its having been replaced by something better.
I'd say that ‘old’ is neutral, for the reasons that Rippler gives. (Some people devalue old things, but that's on them, not the word.) But if you wants to indicate value, then you can use ‘antique’ and ‘antiquated’, as already stated. (But all of these words can be given different connotations by context.)
Fine. One final counter-example: an old town.
A town is never "antique", and people feel great attachment to a town that is "old". Sure most of these examples are in context, but you can't deny that people do in fact use old in a positive way. "Old-fashioned" is even often used to mean "better than the new stuff": old fashioned donuts, old-fashioned soda, it's a common marketing technique.
And finally, people very often simply say, "he's an old friend" (e.g. Han Solo when referring to Lando Calrissian in The Empire Strikes Back) and the use of the word "mine" means nothing, since that is used when referring to enemies as well.
There is usually the word "dear" or "good" or at least the possessive "my", but even without extra words, the word "friend" is such a positive word that it splashes over the word "old". The word old can mean more than one thing. It can mean age, or it can mean you have had or known something or someone a long time. "That is an old house." does not have the positive connotation that "That is my old house." has. You can, however, get away with "He is an old friend.", because people assume you mean "He is an old friend of mine." or depending on the conversation a personal friend of someone that you have been talking about. Also, someone who is an old friend is not necessarily old if you have known them since childhood.
Old is not necessarily worth a lot of money like an antique is. My old house has value to me, because I lived there and have memories of it. It could be ready now to be torn down, but I will always have fond memories of it and maybe I would rather restore it back to its former state, but I could be alone in that feeling. Something that is antique has a recognized value to more people. The value increases for something that is antique, the older it gets, as it becomes rarer because people don't make it that way any more. You may have no personal attachment to something that is antique, except that you know a collector will pay big money for it.