I was looking for an article about something else and ran across this. I mentioned this in a post a couple weeks ago. This is just the introductory summary. I'm pretty sure I found this initially on Google Scholar if you want to search for it.
National Journal of Otorhinolaryngology and Head & Neck Surgery, Vol. 1(10) No. 3, December 2013
Conclusion External trans-cervical approach is better procedure than intra-oral approach in excising the elongated styloid process because of better exposure, less operative time, less blood loss, cost effective, can be operated in high risk patients and greater length of styloid process can be removed.
I ended up having the trans-cervical approach for resection of my styloid process. In discussion with the doctors, one of the main advantages of this approach is the clear window the surgeons can have to the styloid. This allows a good approach with care to avoid any injury to the surrounding neurologic structures. Of course, the downside is the scar on the neck but typically this scar is quite small. I think other users have posted pictures of their scars. Mine was bigger than usual because of other things that had to be taken care of in addition to the styloid process. Another advantage of the external approach is that there's less risk of infection. As you can imagine, intra-oral approaches through the mouth result in a lot of manipulation of the mouth flora which may possibly interfere during wound healing process (I don't have a citation for this at the moment... the risks are very low). This risk is lowered with the external approaches (but yes there's always a slight change of infection with any surgical procedure).