what is the difference between and alternator like on a modern car and a generator like they used on older cars?

Both generate AC current by moving coils of wire near electromagnets, the devil is in the details. A generator spins the current generating coils and has the electromagnets be stationary, where the alternator does the reverse. The upshot of this is that in the alternator, the relatively low current required by the magnets goes through a set of brushes, where in the generator, the high generated current goes through the brushes (which generates more heat, so the brushes wear faster). The generator also does the DC conversion mechanically, by having the ring the brushes rest against be split into bits (the commutator), which only transmits the "positive" end of the AC sine wave, so you get a series of interrupted current pulses only flowing in one direction. The commutator also accelerates brush wear. The alternator, since the current generating coils are fixed, can't use a mechanical commutator, so it uses a set of high-current diodes to cut the negative side of the sine waves instead.

The alternator can generate more current than the generator for a given size mostly because it can be multi-phase. Rather than just one set of current generating coils, an alternator will have several (usually 3), all slightly out of phase with one another due to the rotational angle differences of the magnets and the coils as they spin. These multiple phases all "add up" to more current.

Aren't alternators also full wave rectified as well? (flipping the negative side of the sine wave into positive instead of just cutting it off)

To my knowledge, alternators will charge effectively at idle while generators need rpm to reach full voltage. At least that's the old school theory that they told me during my hot rod days long ago.

and

The generator is long and round. The alternator is short and round