1. Some differences in features.
2. Perceived trustworthiness.
3. Who's there - who can you interact with on each one.
Originally DW started from LJ's code, but both they and LJ have independently made changes, so although the two are still quite similar, each has features (or misfeatures, in some cases) that the other doesn't. Overall, the impression I get is that DW is a little better on the feature front, for people who prefer staying closer to the spirit of what LJ was like. However, I hardly ever hear anyone say that that's why they switched from LiveJournal to Dreamwidth, or using that as the reason to urge others to switch. Almost universally, people allude to #2.
What it boils down to is that LiveJournal was originally well trusted, but then it sold to less trusted owners. Dreamwidth's founders, as far as I can tell, aren't seen as better than original LiveJournal; some people are just more comfortable with them than with LJ's current owners. But the very same thing could happen to Dreamwidth: they, too, could sell to less trusted owners.
So it it worth the time and disruption of switching over to something that may be as good as what LJ used to be, but could later become what LJ is now? Which brings us to #3 - LiveJournal is still where most of the people are. Which means that, on balance, LiveJournal remains the superior service. Feature differences aren't that huge, so they don't outweigh the fact that far more of the people I want to interact with are here compared to there.
Originally, Dreamwidth made a big deal of their founding documents as supposedly a basis for trusting that Dreamwidth won't sell out in the future like LiveJournal did. It makes a lot of sense for them to have done that, because that would've been the main reason for founding a LiveJournal alternative. But I think they botched it: I read those documents, and as far as I could tell, the key difference was that LiveJournal had been subject to one person's whim to sell, while Dreamwidth is subject to two people. I guess that's a bit better, but it's no security.
Worse, when I went to the Dreamwidth IRC channel back when the project was first announced, to try to confirm my interpretation of the document... wow, were people there nasty and mean-spirited and defensive to the extreme. By asking some factual questions in several different ways, I did eventually succeed in confirming that I'd interpreted the document correctly, but people involved in the project seeme to universally view such questioning as personal attacks in the intentions of Dreamwidth's founders, and responded with hostility and insults. That experience left a bitter taste, and a gut impression on my part that Dreamwidth is actually less to be trusted than LiveJournal.
I've got an account there in case there's ever a mass migration from LJ to DW, to make it easier for me to follow my friends there should it become necessary. But if you're curious why I'm not at all interested in supporting or instigating such a thing so far, now you know.