In the simple Hückel method (SHM) the basis set is limited to p orbitals. This set is limited in a great extent to pz orbitals which constraints the molecular plane to be the xy plane. Basically, you are limited to planar molecules.
The inclusion of all valence s and p orbitals in the extended Hückel method (EHM) naturally lifts the spacial constraints and you can work with non-planar molecules.
These are the two first differences between the two methods. The other points are:
- Orbital energies are limited to same-atom interactions, adjacent-atom interactions while all other interactions are 0.
- Fock matrix elements are not actually calculated.
- Overlap integrals are limited to 1 or 0.
- Orbital energies are calculated and vary smoothly with geometry.
- Fock matrix elements are actually calculated.
- Overlap integrals are actually calculated.
You can look up the derivation and steps for the implementation of these two methods in this book that I used as a reference:
Errol G. Lewars; Computational Chemistry, Introduction to the Theory and Applications of Molecular and Quantum Mechanics, Second Edition;
Springer: 2011. DOI: 10.1007/978-90-481-3862-3