Forward drop of diode vs forward drop of LED
Different semiconductor junctions have different forward voltages (and reverse leakage currents, and reverse breakdown voltages, etc.)  The forward drop of a typical small-signal silicon diode is around 0.7 volts.  Same thing only germanium, around 0.3V.  The forward drop of a PIN (p-type, intrinsic, n-type) power diode like a 1N4004 is more like a volt or more.  The forward drop of a typical 1A power Schottky is something like 0.3V at low currents, higher for their design working currents.

Band gap has a lot to do with it -- germanium has a lower band gap than silicon, which has a lower band gap than GaAs or other LED materials.  Silicon carbide has a higher band gap yet, and silicon carbide Schottky diodes have forward drops of something like 2V (check my number on that).

Aside from band gap, the doping profile of the junction has a lot to do with it, too -- a Schottky diode is an extreme example, but a PIN diode will generally have a higher forward drop (and reverse breakdown voltage) than a PN junction.  LED forward drops range from about 1.5V for red LEDs to 3 for blue -- this makes sense because the LED mechanism is basically to generate one photon per electron, so the forward drop in volts has to be equal to or more than the energy of the emitted photons in electron-volts.