I could never be happy with a blog anywhere else. I got my Journal in the time when the invite code system was still in effect. I bought a paid account from the start because I didn't know anyone who had a Journal who could hook me up with a code. After I got my Journal, people got clued in on that, which led to quite a few people I know getting Journals as well.
So what's cool about LiveJournal? Allow me to elaborate.
1. LiveJournal gives you control
LiveJournal gives me control about who gets to read what. Anyone can read your Blogspot blog, which necessarily means you are censoring yourself. Surely there are aspects of your life that you do not want to share with certain people (such as your parents?). In a public blog, that means you can never write about those issues -- no letting off steam, no seeking support from your online friends.
LiveJournal, by contrast, allows you total control of who sees what. You can post entries only accessible to yourself, your friends, or a subset of your friends. Almost no-one gets to read everything on my Journal -- I have several filters in place to ensure that nobody gets to read things I don't want them to read.
By having control over the audience, I feel free to post about things that I would otherwise not have written about. Sometimes it's something as mundane as spoilers for geocaches we've found, sometimes it's a rant about my work that could get me into trouble if it was leaked.
2. LiveJournal is a social network
LiveJournal is a means of interaction with others. You post something, people respond, other people respond to those responses, etcetera. All in an easy to read, threaded way -- which makes it easier to keep track of conversations.
I think that's the key word: conversations. A LiveJournal allows you to have a conversation with your readers -- if comments are enabled, you can respond and will often get a response back.
Shamus Young wondered why he got so many replies on his blog posts from LJ users. I guess LJ people are more sociable and more used to get involved in the discussion. That's what LJ is about.
Also, LJ has communities.
3. LiveJournal is an aggregator
LiveJournal allows me to add RSS feeds to my friends list. I only have to navigate to my friends page, and I get all the posts I need to read. If I encounter a blog I want to keep up to date with, I just add the RSS. Cool webcomic? Just add the RSS. News site? Add the RSS.
I live in my friends page.
4. LiveJournal is open
The codebase behind LiveJournal is open source. Now, this may not be much of an issue for most of you, but to me it means that there are more eyeballs scoping out bugs, submitting patches and checking security.
There are also a lot of LJ clones out there, all with their own culture. If you want, you could also set up your own LJ clone for your own private network.
5. LiveJournal has a large userbase
LiveJournal has lots of users. Lots of users. With that many users, other organisations are more willing to provide interesting features. For instance, voice posting through nifty VOIP-gateways. Or posting by SMS.
The huge userbase of LJ means that it's interesting for others to offer their services to the LJ audience.
6. LiveJournal doesn't show me ads
LiveJournal did add a Plus account, which offers some features in exchange for showing you ads. But I have a permanent account, and I will never get to see a single ad -- no matter what happens.
All in all, I am very happy with my LiveJournal account. There are some powerful features that I need to get my blog on, that I couldn't get anywhere else.