This is a conversation I have regularly with many people, both in person and online via the various groups I'm involved in. It's a common misconception that the choice just comes down to brand, plus the different bells and whistles that the respective marketing teams decide to focus on at any given time.
However, this is most certainly not the case, and having seen and heard numerous people commenting that 'all modern interchangeable lens cameras are basically the same', I thought I should set the record straight. But first, a little background . . .
History
Mirrorless cameras are relatively new to the photography world, having only been available for around the last decade. Initially they were marketed as 'Compact System Cameras', due to being smaller and lighter than DSLRs, but unfortunately this caused many to think they they were only as good as a traditional compact 'point and shoot' camera, and hence not to be taken seriously. Which didn't really get things off to a good start for the manufacturers!
Main difference
The key difference between mirrorless cameras and DSLR cameras is actually just as the name implies. DSLRs have a mirror system (otherwise known as a pentaprism) which transmits the image from the lens up into the optical viewfinder.
Mirrorless cameras have replaced the optical viewfinder with an electronic viewfinder (EVF) - or dispensed with the viewfinder altogether and just use the screen on the back - meaning they do not need the mirror system. This enables them to be generally smaller and lighter than DSLRs, which is the main advantage that was originally marketed.
Early adoption
For the first few years, mirrorless cameras were not taken seriously. Mainly because the quality of the EVF, when compared with an optical viewfinder in a DSLR, just wasn't good enough - or because they did not have viewfinders at all. However, in the last five years or so, the quality of EVF technology has dramatically improved, to the extent that mirrorless cameras are now more than capable of standing up next to DSLRs. They even offer some extra 'usability' benefits, with cameras like the Lumix G9 for example providing three different size options for the screen within the viewfinder, to enable people who wear glasses to see the whole image more easily.
WYSIWYG
Electronic viewfinders provide a huge range of benefits over optical viewfinders - and 'what you see is what you get' is one of the most valuable. The ability to see exactly how the changes you make to your settings will affect the finished image, even before you press the shutter, is a huge game changer. So if you intentionally under or over expose for example, you'll see that in the EVF, rather than having to take the shot and then review. Likewise, if you're shooting on a wide aperture to create shallow depth of field, again you will see that effect in the viewfinder before you take the shot.
This ability makes mirrorless cameras especially useful for people who are just learning photography for the first time, which is why I always recommend them when people ask which camera is good for a beginner. Added to this, because you can review your images in the EVF, you can see straight away if the shot has come out how you expected, whereas with a DSLR you're relying on the screen on the back - which can be very hard to see clearly in bright light.
Flexible focus points
Another key benefit of a mirrorless camera is the ability to position your focal point anywhere within the viewfinder - vs a DSLR which has a diamond shape of focal points and only enables you to pick one of those.
So if you want to focus on something that's right in the corner for example, your only option is to focus on the subject, then half press the shutter and recompose the shot, which is more time consuming and can cause issues with correct exposure.
It's much easier with a mirrorless to simply position the focal point where you want it - either via the software, by using the touch screen, or via the 'hardware' by using controls on the back of the camera. Some models even have a handy joystick, which makes it very easy to position your focal point exactly where you need it. This example actually shows the LCD screen, as it's impossible to photograph what you see in the viewfinder, but I promise you it looks identical!
Silent and high speed shutter
One other thing to bear in mind is that mirrorless cameras have an electronic shutter as well as a mechanical one, which is completely silent. One reason why wedding and event photographers like them so much. This is ideal not only for event photography, but also for things like street photography - or when travelling and visiting religious sites where shutter noise would be inappropriate.
Electronic shutters also enable you to choose a much higher shutter speed - up to 1/32,000 in some cases - vs most DSLRs which don't go above 1/4,000. This is great for sports photographers - who also benefit from faster burst shooting, and from cool features such as the 4K burst mode offered by Panasonic Lumix cameras. A clever trick which enables you to take a 'burst' of 30 frames per second, and then pick the one you like best to save as an 8 megapixel jpeg file - even starting a second before you press the shutter.
Live histogram
If you're used to reviewing the histogram on an image you've taken, to make sure the highlights and shadows are not 'clipped', then an EVF takes it one step further - giving you the option to have a live histogram actually visible in the viewfinder while taking the shot. You can also move it about the screen, to make sure it's not in the way.
This means you can see, even before you press the shutter, whether your exposure is spot on - and hence if you need to make any adjustments before you take the shot. As above, this example actually shows the LCD screen, as it's not possible to photograph what you see in the viewfinder. You'll see I intentionally under exposed, and the histogram clearly shows that - but with many shots it's not as obvious, especially in the case of clipped highlights.
In summary
Hopefully the above explains a little about the differences between mirrorless and DSLR, and why mirrorless cameras offer so many advantages - beyond just being smaller and lighter. If you're a beginner and looking for your first camera, then definitely don't assume that a DSLR is the only option. I'm speaking from personal experience here as I also teach photography. You may also find this article interesting, which I wrote after teaching two people at the same time, one of whom had a DSLR and one that was using a mirrorless. It was an interesting comparison!