I've only flown in a VE - but after having seen several LongEZ's close up, my impressions are:
1) Well the biggest difference is Range. The LE can take you 1000 - 1500 nm at a stretch; while the VE is good for half that distance.
2) The stock LongEZ is about 1.5 to 2 inches wider in the cockpit. I'm 6'1" and 200 lbs, so that extra room is good for my elbows... But I know of at least 1 VE out there that was widened to the LongEZ width (during construction) no problem.
3) The "strake pockets" and overall size of the LongEZ make for more storage... With two guys my size in a VariEze (the configuration I flew it in), there was really no storage space other than a couple of small map-pockets. If you're planning on taking some clothes and gear with you on trips, the LongEZ is easier to pack.
4) The LongEZ with an O-235 or O-320 has a bit more power than a VariEze - but you do have higher costs due to the bigger/heavier engine.
5) The VariEze that I flew in was pretty much a "day VFR only" plane. And I can't see doing much IFR or IMC flying in one (and IMC accident rates in EZ's prove that its not a good idea)... The LongEZ can handle "soft" IFR much much better, by all accounts.
6) Between the engine-power, the panel-space, and the range - the LongEZ is the better plane for long-distance trips. If you're only going to fly recreationally on the weekends around your local area, the Vari is lighter and has a slightly lower cost of ownership.
7) The VariEze is, as it was put, a little sports car. It REALLY reminded me of the spry handling and responsiveness of my Mazda Miata (little 2-seat convertible). I'm a computer-gaming-addict, so my hand-eye coordination is very sensitive... But some folks claim that the VariEze is a little bit "twitchy" for them. I love it, but the LongEZ is reportedly a slightly more stable platform. Not that the VariEze is UNstable - but the LongEZ supposedly requires a bit firmer controlling and feels more stately in its response. I'm betting its still a lot more fun & more maneuverable than a Cessna, though. ;-)

Due to the wing-sweep, the VariEze has a higher touchdown speed, IIRC. Its lighter weight means that both have about the same T/O & roll-out distances - but there's a 5 - 10 knot difference in landing speeds (again IIRC; someone correct me if I'm wrong).
9) Some VariEze's have been hit with some metal corrosion problems - exacerbated by the wing-attach design of the plane. Its not a bad design; just not as convenient as the LongEZ. This issue is really a small matter - just something to be aware of when looking to buy a used plane.
...The BEST advice anyone can give, is that you look for some local canard flyers in your area. Most are very friendly folks - and would be happy to meet with you and show off their planes. Nothing will compare to you taking a close-up look in person, and chatting face-to-face with an experienced pilot.... maybe even sitting in the plane or taking a ride from the back seat, if you're lucky! After you've done this, THEN you'll be able to judge for yourself which plane suits you best.
Take care,
--Noel