What are the disadvantages of using a diode to drop Vcc by a fixed amount?
The main disadvantage is uncertainty about the output voltage.

First, the diode forward voltage may vary over temperature. For example, the jelly bean 1N4004 is likely to change its forward voltage by 100 mV as it heats up from 25 C to 100 C.

Second, if the load circuit has a very low power sleep mode (as many battery powered circuits do) the diode drop could be well below the nominal 0.55 to 0.7 V drop we normally assume.

Here's the I-V curve over temperature for another common part, 1N4148:



If this were used to drop voltage in a device with a 50 mA operating mode and 0.1 mA sleep mode, with operating temperature range from 0 to 85 C, the drop across this diode could range from about 0.375 to 0.9 V, which is a wider range than the allowable Vdd range for many chips.

Many circuits wouldn't be adversely affected by this power voltage uncertainty. But many others would. And in any case it is much simpler to design with the guaranteed 1% accuracy available from linear regulators than to worry about 100 mV or more of uncertainty given by the diode solution (but also watch out for linear regulators not dealing well with micro-power loads).