If you use it, what are the advantages of traditional grip? I've seen the grip explained in various beginning mallet books, but I have never personally seen anyone use it. Why would traditional grip players prefer it over another cross grip such as the burton grip?

Hello,
I am a traditional grip user. I went for about six months playing Burton, but realized that it is very limited, in my opinion. Burton grip is good for vibe playing, because the vibe reportoire (whether it is jazz improv, or compositions for vibes) requires there to (usually, but not always) have two notes on the left hand playing chords, and then a primary melody. The melody has to come out more, so Burton is powerful in that sense. The inside mallet on your right hand (when not used) is supposed to be help horizontally near your waist, while the outside mallet plays the melody. When playing Burton, you are required to play two-mallet melodies with the outside right hand mallet and inside left hand mallet, which you can imagine makes things kind of tricky, and limits you, if you need to play a higher note after a two note passage.
With traditional, everything is pretty symmetric, so if you have a two note melody, you can play it with the inside mallets of the right and left hand. Another great thing is that playing one handed rolls seem much more natural with traditional. If you do one handed rolls with Burton, then you are prone to have your mallets make a "click-click"-ing sound. The reason why is because with Burton your inside mallet is on the bottom (if your palm is facing down, your playing position), but for traditional its on the top, which somehow makes it work better. Also, because of the mallet being on the bottom for Burton, when doing one handed rolls or just even playing both mallets at the same time, you have to keep your wrist much flatter (kind of tensing, like if you rotate your wrist inwards towards your inside mallet), which makes your muscles in your forearm tense up.
I hope this info helps. I should say though, that these are my PERSONAL feelings and observations, and you may find yourself wanting to play Burton, but at the least these are just some tips to keep in mind.
~Behzad