This is an interesting question. I teach both GRE and GMAT for The Princeton Review, so I have a lot of experience with both kinds of students. Generally speaking, the GMAT is a more rigorous test on the Quant section, and just different on the Verbal. Most people I have had contact with find the GMAT Verbal to be much easier than the GRE Verbal. Essentially, the Verbal on the GRE is mainly vocabulary based, whereas the Verbal on the GMAT is focused on logic and grammar.
If you find that you have a weak vocab and a strong set of math skills, the GMAT is probably your best bet. Conversely, if you have a strong vocabulary and a deficiency in math, you would want to take the GRE.
Now that suggestion is dependent on the idea that your schools have no preference. In other words, more to the point of the previous poster, it is possible that some of your schools will not know exactly how to interpret your GRE score in comparison to the other GMAT taking applicants.
If you do decide to work with the GMAT, I advise taking some prep courses to help your transition. My company offers a GMAT course and is currently discounting the course by 15% if you sign up through GMATClub (the offer is under the "partners" forum.) At the very least, a prep course is helpful to get a comprehensive understanding of the exam. There are also a lot of supplementary materials you can use for your prep that are invaluable, such as the
OG and GMATPrep software.
If you have any questions about the tests themselves, feel free to PM me! I am instructor at The Princeton Review for both exams, and a trainer for the GMAT, so I have a pretty good understanding of the tests and the students who take them.