Symmetric matrix is always diagonalizable?

Diagonalizable doesn't mean it has distinct eigenvalues. Think about the identity matrix, it is diagonaliable (already diagonal, but same eigenvalues. But the converse is true, every matrix with distinct eigenvalues can be diagonalized.

It is definitively NOT true that a diagonalizable matrix has all distinct eigenvalues--take the identity matrix. This is sufficient, but not necessary. There is no contradiction here.

There is a difference between algebraic multiplicity of eigenvalues (how many times an eigenvalue shows up in the characteristic polynomial) and their geometric multiplicity (nullity). I presume this is the confusing part!

For diagonalizability, it is required that geometric multiplicity to be equal to the algebraic multiplicity of eigenvalues.

See similarity transformation for more details.