To answer your question first we need to know why do we need quantum mechanics in thermodynamics:
In Quantum mechanics you can attribute a wave function(to be precise wave-packet) to a particle. .
At high temperatures particles can be pictured as billiard balls because their size is much smaller compared to interparticle distance. But as the gas cools down the particles get closer and closer and their wave packets overlap with each other
"De Broglie Wavelength" is the wavelength of these particle waves. In the limit "Thermal Wavelength"(a constant multiplied by De Broglie Wavelength) is smaller than the interparticle distance we are in the classical regime but if the Thermal wavelength is comparable or longer than interparticle distance then we need to consider quantum mechanical effects.
Now that we know that at lower temperatures we need to use quantum mechanics instead of classical mechanics we get back to your question. The fundamental difference that leads to different and interesting phenomenon in quantum statistics is that particles in quantum mechanical system are indistinguishable. Meaning if you swap two particles the Hamiltonian of the system doesn't change. which yields the wave function describing the system has to be either symmetric or antisymmetric under interchanging particles.(for a proof look at P.104 of Introduction to Statistical Physics by Kerson Huang)
If the wave function is symmetric we call the system "Bose Gas" and if it's antisymmetric we call it's a "Fermi Gas". In Bose gas two or more particles can occupy same state, While in Fermi gas each state is either empty or occupied by one particle. This leads to so different behaviour between Bose gas and Fermi gas. For example electrons in solids can be thought of as Fermi gas and Photons as Bose gas.
The quantum mechanical behaviours are so different with what you expect from classical thermodynamics but in the high temperature limit one can show that both Fermi and Bose gases approach to classical statistics and you get the same classical thermodynamic.
One last interesting example is that in classical thermodynamics we can find the entropy of gas up to a constant but in quantum statistics we can find the exact constants in entropy formula so in that sense it gives more information than classical thermodynamics.