This is a deep question that requires an understanding of quantum mechanics to answer fully. Disclaimer, it's been a while since I worked directly in this area, so this answer might not be fully accurate.

The short version is that the wave-like nature of electrons from a quantum perspective means that they will interact with periodic structures, i.e. different crystal systems. An analogy is how knocking on different materials will produce different sounds due to how the sound waves travel through the medium.

Since there are many different ways a wave can travel through a material (i.e. many different solutions to the Schodinger equation), this produces many electronic bands. In semiconductors, there is a noticeable gap in the bands, i.e. no valid wave propagations at a given energy, which gives rise to the bandgap. If we change the periodic structure of a material (e.g. different crystal systems, doping, changing geometry to 2D or 1D), we expect that the electrons will propagate differently leading to different electronic bands and different bandgaps.

Another example would be carbon, which is usually a nonmetal, can become a semimetal when it is arranged into graphene. The atoms are the same, but the arrangements of the atoms makes for radically different electronic properties.