That Nikon is a dSLR (digital single lens reflex). The dSLR design uses a mirror and a pentaprism to send the image coming through the lens through an optical viewfinder. That adds mechanical complexity, physical depth, and lots of moving parts to the camera body. However, SLRs have been around since the late 1950s, and are well established. The finder goes dark at the moment of exposure, just as it does with your Canon FTb. The mirror has to flop up and slam against a foam dampener. Then the shutter opens and closes. Finally, the mirror comes down. That can add vibration and introduce lens-to-body alignment issues that result in focusing errors with some lenses. Mirrorless cameras use the sensor to focus directly, so there are no moving parts to be aligned. But the technology used to focus mirrorless cameras can be slower than that used in dSLRs.
The Fujifilm is a MILC (mirrorless interchangeable lens camera). Mirrorless cameras use the sensor to send an image to an electronic viewfinder or to the rear LCD/OLED screen. You see an image that is processed through the camera electronics, so you can adjust menu settings, exposure, etc. and see the effects in real time. You know what you are getting before you get it. The down side of that, thus far, is that the processing takes time, introducing a very tiny delay between what actually happened and what you see. That can be critical for some applications, and insignificant for others.
Mirrorless bodies can be slightly lighter than dSLRs. But APS-C format Fujinon lenses and DX Nikon lenses are roughly equal in size and weight for the same focal lengths and maximum apertures. If you want real weight savings from a mirrorless camera SYSTEM, you need an Olympus or Panasonic Lumix Micro 4/3 system.
Fujifilm cameras have a nice set of buttons, dials, and knobs on them. If that is what you are used to, you may find that camera much easier to use. The menus on that camera are also very straightforward and easy to use. The D5600 Nikon is a fairly well-refined design, but has its own style. I would urge you to find some avenue where you may try BOTH cameras to see which makes more sense in your hands.
Nikon is the #3 camera brand in total sales, behind Canon (#1) and Sony (#2). Fujifilm is down the list a bit, but they make EXCELLENT bodies and lenses. Also available are excellent cameras from Olympus and Panasonic and Pentax. Another camera I would seriously consider, since you like the Fujifilm X-T20, is the Panasonic Lumix G85. It, too, has plenty of real buttons, dials, simple menus, and an ergonomically appealing design. The Sony a6300 and a6500 are also worth considering, as is the Olympus OM‑D E‑M5. But they are not as easy to use for some people (the menus can be hard to navigate).
Of all these, my picks would be the Fujifilm X-T20 and the Lumix G85. Final decision on those would come down to sensor size vs. system weight.
You can find reviews of all these cameras at
http://www.dpreview.com, and all over YouTube.
I really don't think you can go wrong with any choice in this class. You might consider whether a friend or family member has compatible lenses you might borrow.
That Nikon is a dSLR (digital single lens reflex).... (