Some of the sites where I'm most active are sites in public beta: from my network profile, my 2nd, 3rd and 4th most active sites (by reputation) are Travel, Academia and The Great Outdoors, respectively. In my way of asking questions and giving answers, should I care if a site is in beta? Is there anything I might do differently on a beta site as opposed to a promoted site?
You should be prepared to "roll with" a tweaking of the site's scope. Some questions that were on topic when the site first went into beta might not be when it is ready to graduate. As a beta user, you should know that and be ok with it. This might include editing really old questions that bump onto the main page, or it might just mean not asking your own variant on something that isn't on topic any more. It helps to read the per-site meta once in a while to get a handle on the shifting sands, if any exist. Of course, you can always join in on this process if it hasn't finished happening, and help to shape a site that you will like to use.
You should help the site to grow. This might include tweeting links to questions or telling your friends about the site. Beta sites really need this help. It might also include answering, upvoting, or commenting nicely on something that might not normally attract those efforts from you, as a way to reward someone for asking a question and contributing to the site. It might include asking a question (one whose answer you want) a little sooner than you might have, because you notice there hasn't been a new question for a day or two and you don't want the traffic to fall off. You might even ask a question whose answer you already know, in order to get a high quality on topic question onto the site that will help future visitors.
You should be prepared to take on semi-mod duties sooner than you would on a non-beta site, and to take those responsibilities seriously. You're about to gain full edit privileges on Travel, for example, at half the rep it would normally take. That's because the smaller user base makes it tough to get enough help at the rep levels graduated sites use. Take what you know from using other sites in the network and use it to make this one better.
I think those three are the big ones.
Every beta have some risk of not passing the test and then disappear, leaving you with absolutely zero privileges, badges and reputation you have earned.
Everything would be lost if not because they are kind enough to provide a read-only backup of the content with all the Q&As, while their intentions is good, the main purpose of this copy of the site is to make you more sad by immortalizing what you have lost.
So, yes, you should care if it is a beta.