1. There's supposedly a space/weight saving by not having to have a mirror and pentaprism to give you a viewfinder. Unfortunately, this space/weight saving seems to mostly have been negated by the tendency to make the lenses bigger and heavier.
2. The viewfinder is electronic, not real. This may take some getting used to. It may also be less practical in some situations and more practical in others. Consider when there's loads of light around (you're wearing sunglasses and the electronic viewfinder is too dim) or when it's really dark (the DSLR viewfinder is too dark). I'm told that the electronic viewfinders these days are really good, but I'm sure it's a matter of preference.
3. There's no mirror that flips up, which can reduce camera shake. You can even get most of the mirrorless cameras to use a fully-electronic shutter, so taking a picture can shake-less and silent.
4. The lens mount is closer to the sensor, which can make it easier to make some lenses (mostly ultra-wide angle lenses).
5. The focusing system is usually way better. DSLRs depend on a translucent part of the mirror to direct some light onto a focusing sensor. Mirrorless cameras have a load of split pixels scattered across the main sensor, that allow them to do phase-shift focusing. But because they're using the main sensor, they can do things like work out where people's faces (or even eyes) are, and focus on that.
6. The battery usually doesn't last as long as with a DSLR. For example, I went out one night and took 1758 photos on a single battery on my DSLR. You'd need more than one battery on a mirrorless camera to do that.
I might be wrong, but I thought most DSLRs use phase detection, while mirrorless cameras use contrast detection. Phase detection tends to be much faster, but contrast detection is more accurate.
There is no mirror! You know, the fragile, fast-moving, complicated part that upsets the camera at the very moment it needs to stay still.
Shooting portraits with a z6 and a 60 year old 90mm leica lens on a megadap autofocus adapter (it just moves the whole lens in and own) yields some really cool images.
No mirror means easier and smaller packaging due to not having to make space for the mirrorbox and prism.
For existing customers (like you), the disadvantage is that they may have to throw their kit away and start over, for some advantage in weight and in body technology, but need to pay a lot of money too.
The Z5 is better for baby pictures.
The d5600 is still better for hiking and travel pictures.
The Z5 is better under low-light and dynamic subject conditions. Particularly because the digital viewfinder is clearer in the dark, the eye/face autofocus and low light autofocus performance is much better (much fewer missed shots), the Z5 has sensor-shift stabilization, and there is no mirror slap affecting lower shutter speeds. For landscapes where there is lots of light, a lightweight and battery conservative d5600 still does the job excellently.