At least from their names, it seems motional emf is induced, so what's the difference?
Physically, they're the same, but historically this was not understood until special relativity.
We have a loop of wire and a bar magnet. In one experiment, we move the magnet towards the loop of wire. The flux through the loop increases. The loop now has a current, meaning there is an electric field pointing around the loop. This is induced EMF. Induced EMF is calculated using the change in flux through a stationary loop.
In a second experiment, we hold the magnet still and move the loop towards it. We start the same distance away and move the same speed, but change the role of stationary and moving agents. Now, $\partial B/\partial t = 0$ because the magnetic field is not changing, so there is no induction. Instead, there is a Lorentz force on the charge carriers in the loop because they are moving through a magnetic field. The induced current works out to be exactly the same as before, though.
Physically, the same thing is happening, but the mathematical description is different depending on whether you view it from the rest frame of the magnet or the rest frame of the loop. One of the most important physics papers ever, Einstein's "On the Electrodynamics of Moving Bodies", begins by pointing out this strange coincidence.
It is known that Maxwell's electrodynamics--as usually understood at the present time--when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena. Take, for example, the reciprocal electrodynamic action of a magnet and a conductor. The observable phenomenon here depends only on the relative motion of the conductor and the magnet, whereas the customary view draws a sharp distinction between the two cases in which either the one or the other of these bodies is in motion. For if the magnet is in motion and the conductor at rest, there arises in the neighbourhood of the magnet an electric field with a certain definite energy, producing a current at the places where parts of the conductor are situated. But if the magnet is stationary and the conductor in motion, no electric field arises in the neighbourhood of the magnet. In the conductor, however, we find an electromotive force, to which in itself there is no corresponding energy, but which gives rise--assuming equality of relative motion in the two cases discussed--to electric currents of the same path and intensity as those produced by the electric forces in the former case.
Special relativity allows us to understand this problem by describing how different observers perceive magnetic and electric fields when they are in motion relative to each other. If you walk towards me down a hallway, we will measure different electric and magnetic fields in the hallway, even when you're right next to me as you walk past. Once we understand how to transform the magnetic field of the bar magnet, we see that induced and motional emf are just transformed versions of the same thing, accurately described by special relativity.
Induced emf is equivalent to a bar magnet being brought closer to/moved farther from the loop. And motional emf is equivalent to bringing a loop into /moving it farther from the magnetic field.