I took the GRE in the fall (Q:800 V:580) and am taking the GMAT in 2 weeks. One thing is for sure, you really cannot say one is "easier" than the other. Although there are some similarities, they test many different areas. The verbal section in the GRE is basically a vocabulary test and as my score might show, my vocabulary is not the greatest. This doesn't mean my english is weak (I came to the US to attend undergrad and was placed in the English writing class for native English speakers and not the one for international students that most of my intl friends took). I just have a lousy memory which explains my low vocab skills. (I also must mention that my Verbal score was lower than even my expectations because I was so drained by the time that section came around). Anyway, the GMAT verbal section is much less focused on vocab and more on actual English comprehension and sentence construction skills which works great for me (scoring >90 percentile in practice tests so far). As far as quant goes, the GMAT does throw out more challenging questions than the GRE and overall I find the math questions on GMAT tougher than on the GRE. However, that doesn't mean the GRE is easier. If you're looking to get in to a top tier school in a quant focused field, you pretty much need to get 800 to give yourself a good chance. There's that much pressure to get the 800 since anything below that makes you less competitive. What I mean to say is that the GMAT offers you a little more wiggle room, anything above 700 is considered good. Get anything below 780 on the GRE quant and you can forget about that econ phd from MIT (in general).
If your school/program of choice accepts either, I say figure out where your skills are and which test works better. Maybe try doing a practice test for both and see which one you like better. One thing I've learnt though is that having done the GRE recently has really helped me get ready for the GMAT so if you want to take your chances at both it wouldn't add much additional effort - just like doing a dual degree n saving a couple of years of time than if you had done them separately.