I think along with your relationship to your professor, it also depends on the professor's personality as well. For example, asking in person may be best if the professor is old school and would appreciate the effort to come in person, if he or she is not very good responding to emails, etc. Asking through email may be best if the professor is very shy, always busy etc. Personally, I asked in person and then followed up with an email.
Especially if you are applying to Speech Path, I think it's also important to consider asking a professor who is/was a Speech Pathologist or Audiologist, since they can comment best on how you will be as a clinician. Some schools actually require this, so I recommend checking into your programs!
Hope this helps