A list of things (largely technical features) that are being done differently (note: there are many things on that list that aren't covered in my run-down; go check it out if you haven't already.)
Similarity: LiveJournal and Dreamwidth are both social media blogging services, having the ability to publish both single-user and communal blogs, with exceedingly granular security features on each entry, a fairly powerful aggregator, with some social networking features.
Difference: LiveJournal was originally conceived and developed by a hobbyist in a dorm room. Dreamwidth was conceived and developed by owners who had experience with running a social media site.
Difference: LiveJournal has a large company owning it, based out of Russia, although LJ's local stuff is handled by the US-based LJ Inc. Dreamwidth is owned by two individuals, and is run out of Maryland.
Similarity: Much of the technical support for both sites is crowdsourced, and any user on either site is welcome to provide technical support for other users.
Difference: LiveJournal's technical support uses a very formal style, which can be seen positively as professional, or negatively as robotic. Dreamwidth's technical support uses a more natural style, which can be seen positively as human and accessible, or negatively as too casual.
Difference: LiveJournal's Frequently Asked Questions style has a level of formality similar to the support writing style, and FAQ entries are lengthy, containing a large amount of information about each feature, collected into one place. Dreamwidth's Frequently Asked Questions style is more casual, and FAQ entries are shorter, with a larger number of different entries on similar subjects. (Dreamwidth's FAQ is also still under heavy construction, while LiveJournal's FAQs are largely completed, but are updated periodically.)
Difference: LiveJournal is funded both by paid accounts purchased by users and by funds from advertisers. Dreamwidth is funded by paid accounts purchased by users.
Difference: LiveJournal's permanent accounts have extra features above and beyond what users making recurring payments may purchase. Dreamwidth's "seed" accounts have the same features as Premium Paid accounts.
Similarity: Any sale of a one-time payment account that is good for the life of the service trades the possibility that the owner of the account would continue to pay past the depreciation point of the one-time payment for the certainty of cash in hand now.
Difference: LiveJournal does not discount the possibility that there will be another permanent account sale in the future. Barring unforeseen circumstances, Dreamwidth explicitly plans that there will be no more seed account sales in the future.
Difference: LiveJournal's architecture encourages a unwritten social contract that to read meant to allow informational intimacy, and vice versa, even though there were existing methods to avoid this. Dreamwidth's architecture breaks this link, and it is yet to be seen what social norms will evolve around the reading/access model. [Edited for clarity: LiveJournal is built around the 'friends' model, which includes both access and reading. Even though you can evade that with custom friends groups (for both reading and post security) the assumption is built into the service, and therefore into the culture.]
Similarity: Both LiveJournal and Dreamwidth are open-source software projects, to greater or lesser degrees.
Difference: LiveJournal's code is difficult to install and use out of the box, and several important functions are missing from the Open Source portions of the site. Dreamwidth's software is not yet all the way in the box, but is intended to be usable with only the Open Source bits.
Difference: LiveJournal is running production code, with the occasional small exception when it allows its Support volunteers to beta-test and find the inevitable errors in new code before releasing it into production. Dreamwidth is running beta code for all users, who are finding the inevitable errors in the new code.
Similarity: Both LiveJournal and Dreamwidth are for-profit businesses.
Similarity: Both LiveJournal and Dreamwidth are aimed at a general audience, LiveJournal implicitly by not using exclusive language, and Dreamwidth explicitly, by using inclusive language.
Difference: LiveJournal chooses, when such is brought to its attention, to impose further restrictions beyond those required by law upon the content on its servers, in part for keeping the 'www' subdomain at least at R-rated or under, ideally not over PG-14. Dreamwidth's policies deprecate restrictions other than those required by local law and sanity in the name of keeping things spam-free.
Difference: LiveJournal has operated under several different owners in its online tenure. Dreamwidth's ownership has not changed in its brief lifespan so far.
Difference: LiveJournal is a well-established service with history and reputation, and has managed to stay in existence now just over ten years. Dreamwidth is still in (open) beta testing, has only a few months of actual, active, non-hypothetical existence, and has only the reputation of its various and sundry members, supporters, and detractors.
Difference: LiveJournal has an official mascot, Frank the Goat, which is either pretty cool, or actually kind of cheesy, depending on your perspective (and how much you like chèvre). Dreamwidth has no official mascot, although there is some userbase support of sheep and paper bags; this is either refreshing or a bit disappointing depending on how much you like sheep (and paper bags).
Difference: LiveJournal, starting from the ground up, had to invent many of the tools it then used, without the luxury of being able to wait to design it properly. This led to much innovation, and also a certain number of shambling horrors of the deep. Dreamwidth inherited a certain number of shambling horrors of the deep, which it has been merrily ripping out and replacing, having the advantage of hindsight.
Complete Unknown: Your friends may be on Livejournal, on Dreamwidth, or both. They may be cross-posting, posting one place and directing comments onto the other, posting different things both places, or not posting anything at all.
There are more. There are always more. But that's a bit of what I've seen so far.