Why does the resistance of thermistors made from semiconductors decrease with temperature, while the resistance of metals increases?

Because conductivity requires free electrons to participate in conduction. In semiconductors this can only occur when the outer electrons have enough energy to jump into the conduction band. This requires thermal energy at least the size of the band gap. This is typically a few electron volts (eV). Around a band gap of 5eV a material becomes an insulator. The higher temperature of the semiconductor provides this energy and so leads to more electrons and thus lower resistance.
In metals, there are always free electrons because a metal's Fermi level is in a conduction band to begin with. But as the electrons flow they impart energy to the crystal lattice (the stationary atoms of the metal) by bumping into them. This is what electrical resistance is. The hotter the metal the more the lattice is vibrating. This means electrons are more likely to bump into them, thereby increasing the resistance.

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