I used to think of excuses as lame cop-outs, but I've come to the conclusion recently that they are actually much worse than that. Excuses are lies. You may be giving a truthful statement in your excuse, but the implication of an excuse is "I want(ed) to but there is(/was) no possible way because..." The reality is, where there is a will, there is a way. Imagine if your life, and all the lives of those you cared about, relied on you doing that one thing, and if you succeeded in doing that one thing, you would be rewarded with millions of dollars. Could you have found a way around the barrier under those circumstances? If the answer is yes, then making an excuse is lying. The truth, in that situation, is that you're not willing to; the perceived cost/benefit is not great enough for you to work around the barrier.
Certainly, there are circumstances where the cost is obviously much higher than the benefit, and that could be easily discussed, like "I didn't come to work yesterday because I had a heart attack and I was in the hospital being treated." Very few people would argue that coming to work was more important than being treated for a heart attack. Most excuses are much less dramatic, like, "I have to do the laundry so I can't make it to dinner tonight," and those are the types of excuses I just can't stand. If the person you are blowing off is important to you, then I'd say it is more respectful to tell the truth, and show some willingness to compromise. So rather than making an excuse, a truthful statement would be something like, "The logistics of it would be complicated for these reasons, and I'd prefer to do this another time when I don't have so much going on. Can we reschedule? When is the next time you'd be available?"
Perhaps an even more clever way of dealing with it would be to give the option and prioritization to the other person. Present the facts not as excuses but as hurdles that would have to be overcome, and ask them to decide how important it is. Something like, "It would be difficult logistically because of these reasons, but I don't want to blow you off if it's important to you that I do this. Is it that important that we do this right now? Or can it wait for another time?" You may find that the rule of reciprocity plays in here, and the other person may sympathize with your difficulty and decide for themselves that in the grand scheme of things, it is actually not that important. This is not a guaranteed out, though, so only use it if you really are willing to make it work. If the person does say it is important, you may want to ask them why, and that might give you a better perspective on the whole situation; maybe the benefit was greater than you had initially perceived.
Along those lines, can't is a dirty four letter word, in my opinion. To me, the worst thing you could say is, "I can't because..." Conflicts come up all the time, and it's very rarely the case that you actually can't do something, it's more often the case that you've made a decision on what is more important. I think it's much better to say, "I have a prior commitment at that same time which is important to me to keep." Obviously that specific statement is only honest when you really did have a prior commitment that you feel is more important. If you're trying to get out of a prior commitment, then say something like, "An issue has come up that I feel is important to deal with before I..." That's more honest, and it's less of an excuse. I'm generalizing here, but I wouldn't recommend you keep it general with people you care about; give them the specifics of the prior commitment or the issue you have to deal with. People are very sympathetic, and the more information you give them, the more in control of the situation they feel. In a professional setting, I think it's okay to keep it general. But if you keep it general with someone you care about, they may feel shut out completely. If you explain the situation, they are more likely to agree with you that you need to deal with the other thing, and they could potentially even offer help.
Bottom line: when you make an excuse, the implication is that you want to but are being prevented, and the reality is that there is almost always a way to make it work, but you're not actually willing to do that (because the perceived cost/benefit analysis says its not worthwhile). Therefore, excuses are lies to cover up your unwillingness to make the effort. You may be justified, but simply stating an excuse is lying to the other person, it's concealing information pertinent to the situation. It is more honest to put it all out on the table and state your feelings about the situation than to make an excuse; excuses are downright dishonest.