Advantages: The Lojban numbers provide an alternative solfege system. The common solfege system (do, re, mi, etc.) lacks a convenient way to represent accidentals. As you can read in the Wikipedia article, the original system ignored accidentals (this was personally harmful, as I ended up subvocalizing such a solfege; this caused me to have difficulty distinguishing between certain close notes). There is an alternate system, but it is not very well known, and tends to cause an unacceptable amount of rhyming note names in sharp-heavy keys. Also, by naming the notes after numbers, intervals can be easily calculated using mod-12 arithmetic. You can know that the interval between C (0) and F (5), is the same as the interval between F# (6) and B (11).
Thus the beginning of Beethoven's Fur Elise (E-D#-E-D#-E-B-D-C-A) could be sung to by the syllables "vo ci vo ci vo fei re no so".
The system can also easily be adapted to naming notes in systems other than the European 12-note system.
Disadvantages: It doesn't match with the diatonic (major) scale that is usually taught first, and which the traditional musical notation is based on.
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