I ran Canon kit for years. I started with a 350D and moved up to a 5d mark ii (so-called full-frame) with a brace of lenses. A year and a half ago I sold the lot and switched to an Olympus micro four-thirds mirrorless system. I couldn't be happier:
a) It's lighter & smaller
b) It has an electronic viewfinder - a massive advantage for most subjects (fast-action photography (sports etc) being the exception). The best example here is the ability to see the actual exposure before you press the shutter and without the need to look at the shot afterwards ("chimping").
c) It has in-body image stabilisation - which makes a big contribution to sharper hand-held shots
d) It uses contrast-detect AF rather than phase-detect AF. This delivers more accurate and reliable focus. I can also place the focus point practically anywhere in the frame. My 5dii suffered badly from focus consistency but I've hardly had this problem at all with my Olympus gear.
e) Since the sensor size is smaller, the depth of field (what's in acceptable focus) for a given angle of view is deeper - this makes it easier to get all the things you want to be sharp to be sharp without stopping the lens down a lot (which causes slower shutter speeds (camera-shake) and/or higher ISO). This is really useful for people shots up close. Of course the converse is true too - bigger sensors give a shallower depth of field which can be useful for isolating subjects, but for me, this is not a major consideration. I have a great 75mm f1.8 lens for my Oly gear and that gives fantastic isolation. It's also the sharpest lens I've ever used on any system.
f) It has all sorts of interesting tricks courtesy of its advanced "live view" approach - e.g. electronic shutter, fancy long-exposure modes that let you see the image build on the monitor so you can stop it when the shot is correctly exposed, auto-correction of converging verticals (when you point the camera up at building for instance), face (and eye) recognition, ...
g) The biggest interesting trick in the list above, which is a huge advantage if you take a lot of people shots, is face recognition. As an experienced photographer I scoffed at face recognition at first, but I was wrong. Face recognition is the best way to get sharp people shots that are accurately exposed. Combined with in-body image stabilisation and deeper depth of field, I can guarantee that an Olympus camera will give more reliable portraits for average photographers than a typical DSLR.
I could go on. I don't want to overly criticise Nikon/Canon DSLRs (and of course they are hugely popular and give fantastic results), but the truth is that they are a technology that slowly but surely is heading for photography history.
Downsides of mirrorless? - yes, a few:
i) Shorter battery life
ii) Slightly more image noise - becomes more noticeable at higher ISOs
iii) Slightly inferior resolution - but only really an issue if you print VERY large or go pixel peeping. Even experienced critics would be unable to tell the difference between 16" x 20" prints of the same subject made on an Olympus camera and a Nikon or Canon full-frame camera.
iv) Poor AF for moving subjects. If you want to shoot action, then go and get a Canon 5diii or Nikon D4!
OK - too long post. My main message though is this - don't just go for the Nikon/Canon DSLR because that's what everyone else does. Mirrorless cameras are the future and there are increasing numbers of professionals switching. Take a look at this guy's work - he uses Olympus exclusively:
http://www.damianmcgillicuddy.com/general/