As I understand it, TEFL is for teachers overseas (a good idea, but it seems to be geared to feed massive chain language schools like NOVA and whatnot, and those teachers are not known for their high skill) and TOEFL is for ESL students.
I'm not sure but:
TEFL: Teach English as a Foreign Language
TOEFL: Test Of English Fluency Level.
I don't have TEFL, all you need here in Japan is some university, (or not, hey, we are desparate for teachers) and most importantly, real teaching ability. A lot of people teach here in Eikaiwa conversation schools without degrees, but you will never get into a 'real' school without one.
If you can afford a course, it will only make your life easier to take TEFL certification, but its only a beginning.
TEFL is just an acronym for Teaching English as a Foreign Language, rather than a specific course- there are many TEFL qualifications. You would have to take a course to get the qualification. Internationally, the most widely accepted and know is the CELTA.