There are 2 main differences between all mirrorless cameras and all DSLRs, and then a bunch of other stuff that varies between individual models. The 2 main differences are the size and the viewfinder, and the other stuff includes sensor quality, autofocus and lens options, all which are explained in detail below.
1. The Size
Mirrorless cameras are significantly smaller and lighter, because of point 2 below.
But bear in mind if you put a big lens on your tiny mirrorless, you'll find it's top heavy, and you need to put a battery pack on just so you can hold it without pulling all the muscles in your hand, so you start to lose the weight advantage.
2. The Viewfinder
A DSLR (digital single lens reflex) has a periscope between the lens and the viewfinder. The light comes in through the lens, and then has to take a journey to reach your eye that involves a mirror, and then a pentagram.
Why? Because the important bit of the camera is the sensor. This has to be lined up exactly behind the lens to capture exactly what the lens sees. Without the periscope, the viewfinder would just look through the camera and be looking at a slightly higher-up version of the scene than the sensor is seeing. With long distance photos this isn't a problem, but with anything closer you run into the problem of parallax error - you can't judge accurately what will be in the frame.
When you press the shutter, the first bit of the periscope, the mirror, flips up out of the way so that the light passes straight through onto the sensor. It then flips down again once the photo has been taken.
Can you see why mirrorless cameras are called mirrorless now? They don't have the flip up mirror. As technology improved, it was possible to use an electronic viewfinder to beam a "filmed" version of what the lens is seeing direct to your eye. This is either done in a top-of-camera viewfinder, or a back-of-camera LCD display. Either way, you aren't seeing the beam of light you would see with a DSLR, you are seeing an live electronic replay.
3. The other stuff
How does it feel when you hold it?
Go into a camera shop and hold the camera you want to buy, with the lens you want attached. Some people do find mirrorless cameras just too small to grip properly.
Battery life
If you use the LCD on a mirrorless continously, your battery will drain much faster. (This is also true if your DSLR has Live View which you leave on.)
Silent operation
With no mirror to clunk up and down, the default operation for a mirrorless camera is completely silent. Many people like to have the auditory feedback though, so you can turn on a variety of shutter sounds.
Lens options
I'm writing this in August 2017 and there are far less lenses available for mirrorless cameras than there are for Canon/Nikon DSLR systems. But the ones that are available are as good or better quality, and new ones are being added all the time. You can get adaptors if you want to use legacy DSLR lenses with a new mirrorless body.
Sensor size and image quality
Sensors are for the most part smaller than a full-frame pro-spec DSLR, but image quality is outstanding. You can buy full-frame mirrorless cameras if that's important to you. Compare actual prints side by side (camera shops will have these under the counter).
If you will never print bigger than the size you can display in your own home, and you are not a professional sports photographer, in my opinion the image quality will be the same whether you choose a DSLR or a mirrorless (if you are choosing similar priced ones, and using similar quality lenses).
Weatherproofing
There has been a suggestion that mirrorless cameras aren't as weatherproof as DSLRs, but it's mostly anecdotal. I use a Fuji X-T1 in all weathers and have never had any problems with it. I've used it sub zero in Iceland, and in driving rain in Scotland. If you need pro-spec weather sealing you probably know what you're looking for anyway.
What you see is what you get
I absolutely love this feature of mirrorless cameras - being able to see how your camera has exposed, even when you have neutral density filters on. If you use an optical viewfinder on a DSLR with an ND filter, all you see is blackness. With an electronic viewfinder you see the finished image, with all the exposure compensations built in. It also works for white balance errors - you can see straight away if you're off.
Autofocus
If you read reviews from 2016 or before, you'll find scathing reports of the slowness of mirrorless autofocus. It's improving all the time, and my X-T1 is now as fast as any of the Canon DSLRs I've ever owned.
Video
If 4K video is important to you, pick your mirrorless options carefully, and read the reviews.
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