Difference between incandescent/fluorescent/LED wattages?
1) no, 2) yes, 3) almost certainly yes, 4) most of the time yes but see below.

The 60W limit on the lamp is to prevent it overheating and catching fire.  That means that you could, in principle, put a 60W LED in the lamp and it would be OK, or at least the lamp shroud would not catch fire.

An LED must run at a much lower temperature (about 25-50C) than an incandescent (2200-3300C at the filament!), which means it must have much better airflow available to it.

A lamp housing with poor ventilation would probably have no significant bad effect on an incandescent bulb, but it might cause an LED or CFL bulb to overheat and have a very short service life.  So that's one possible reason they would specify incandescent, and note that the lamp you've linked has no way for hot air to escape the top.  Also if there is a dimmer in the lamp or if the lamp contains an SCR-based (solid state) power switch, that would be incompatible with most LED and CFL bulbs.

In the case of that amazon listing though, I think it's just telling you that it comes with an incandescent bulb, not that it is incompatible with LEDs.