Loglan is a predecessor to Lojban, and Lojban apparently directly grew out of Loglan.
How similar are the two languages at this point? Are they more like dialects of the same language, or more like Anglo-Saxon and English at this point?
Lojban vocabulary was deliberately constructed from scratch, in an attempt to get around James Brown's copyright on Loglan (Brown was the creator of Loglan, but continued to retain control over it. The Logical Language Group was formed to reinvent the words, beginning in the late 1980s, while retaining Loglan's grammar.
The words LLG created are the gismu, the roots that make up Lojban. They generated over 1300 of them, taking source words from Chinese, English, Hindi, Spanish, Russian and Arabic and scored many potential gismu by their similarities with the source words. The highest-scoring gismu were used; from there, the rest of the vocabulary was generated.
Now, this was similar to Brown's original process, although he used Japanese, French and German as three more reference languages, while leaving out Arabic. This algorithm was quite similar to the one used to generate Lojban's gisme. Brown's scores - and, in a sense, the LLG's - were explicitly based on learnability, which is why these languages were used. However, the implementation of the Logban algorithm resulted in a completely new lexicon, due to different weightings, quantification of similarity, and the differing source languages.
Since the relexification, the grammar of Lojban has changed in certain ways. While it would have been easier for speakers of Loglan to learn shortly after its inception, it has now diverged enough that the two languages, while based on the same principles, are now very different. I think the analogy of Anglo-Saxon and English is a good one.
Lojban started as a relexification of Loglan but it has evolved independently from Loglan since that starting point. I don't think that the two languages can be considered dialects of each other because there is no mutual understandability left because every single word of Loglan was replaced with something different in Lojban (with the famous exception of the word blanu "blue").