Other answerers pointed out that belts on seats are unfair for standing passengers which can't use them.
Let me explain why trains are intrinsically infinitely safer than road vehicles and airplanes. Train safety is a full branch of rail engineering, and is obviously different from road, air and sea safety.
Please mind reading Eurostat's official statistics for your convenience. (key indicator: 1742 casualties on 2016)
First, note on airplanes: seat belts are not mainly there to save people from full-force crashes (beacause air to ground impacts on full weight are fatal) but mainly to help protect them from turbulence or sudden deceleration during canceled takeoff and landing. Airlines don't want you to hit your head with your front passenger's seat during these events, because insurances don't like to pay damage.
How rail infrastructure prevents crashes
Trains move on a single dimension as they have no ability to steer, so handling their safety is easier than roads. Front impacts and derailments are the only kinds of accidents that rarely occur.
Train crashes are incredibly rare for infrastructural reasons: safe distance is technically enforced by signaling systems. Road vehicles are not compelled to respect a safe distance (for instance, buses in EU do have seat belts and bus crashes do happen on roads), which is also estimated according to travel speed (150km/h - 95mph of max speed for example). Unsafe distance is not simply one of the main crash reason of road vehicles, but also causes the effective crash to happen at faster speed.
At equal speeds and driver's reaction time, of the two cars which suddenly brake the one with longer safe distance will crash at a slower speed than the one tailgating. And tailgating is also a very common phenomena on our roads. We need belts on road vehicles.
Trains are much different in this regard. Consider a 11-cars train launched at 300km/h (220mph?). Not only iron wheels provide a fraction of the grip of tyres on asphalt, the mass of that convoy is infinitely larger than a truck.
Train regulation authorities take this into account and do enforce safe distances by designing rail infrastructure over the concept of
blocks. SNCF (no direct link available as I used other source) estimates a TGV at 300kmh needs 3300m to stop on emergency brake, so the train is reserved always more than 3300m front where no other stock is guaranteed to circulate.
How is that actually enforced? There is no rail police pulling up trains driving too fast or to close to the other but simply the line is electronically controlled so that blocks of predefined size (I recall 1200m for regular traffic 200kmh/130mph and 5400m for high speed over ETCS, see later) are "held" by the train tripping electronic switches.
In the above diagram, each traffic light is separated by
block_length_here of rails. When a train enters a block, its preceeding light turns red and one (2x block), two (3x block) or more (3+x block) change colour according to regulation. Generally speaking, trains are allowed to drive green at max speed, are required to slow on yellow and must absolutely not enter a red because another train is physically driving in that
block_lenght_here space. I have replaced figures with
block_length_here for generality. The above is a general concept and each regulator defines the number of states and effective colours. E.g. a metro service may use only red/green code, or decide to close two blocks rear of the train.
Also, all trains on modern lines are required to equip safety devices that enforce emergency brake as soon as the train passes a red, or yellow too fast.
You can find the above in all modern lines around the world, but consider that the effective signal (circle, square, double-yellow, etc) vary by country, especially in Europe in which each country has its own rail signaling system. But the very concept applies to all.
European Train Control System (ETCS) is an evolution of traditional block-system where there are no more lights and the train auto-regulates its speed according to the exact position of the preceeding convoy rather than on fixed space points. In that case, the driver doesn't even need to slow down as the train does electronically. He can see the distance to the next obstacle/joinpoint/station/whatever on his display
How the design of a train prevents injuries in a crash
It is also interesting to note that trains are also designed themselves to minimize fatal damage in case of front crashes and derailments.
On front crashes, the driver is normally the first victim and hopefully the only casualty, because the majority of trains (I would take Sweden and Denmark immediately out from this list because all their rolling stock have distributed engine power) are pulled by a front engine car, which absorbs a large part of the impact.
Note that the force of impact is not equally distributed to through the length of convoy, but is puposefully designed to dissipate through the front parts. I am simply saying that passengers seating/standing on the middle of the train will be shocked by the deceleration but very unlikely at a fatal force.
About derailments, trains are also designed to limit the number of derailed cars. Consider for example Alstom's AGV trains (slide #20) that feature engine-and-coupler-wheels between two carriages: the manufacturer stated that this design technique, while increasing maintenance costs by not allowing to decouple a car on the rail, allows to consisently recuce the likelihood of a derailed car to flip on its axis.
Train interiors do also have design features to limit the damage to passenger. While a passenger caught by an impact during his way to the toilet will still fall and hit his head on something hard, seated passengers may be (partially) protected by the seat in front of them and/or the table before the reverse seat in front of them. Pick Shinkansen, where every seat always fronts the train's travel direction. Seats are never rigid but allow jostling not only to inclinate them (for traveler's comfort), but also to absorb the impact.