what's the difference between protein synthesis and gene expression?

To put it simply...

Protein synthesis: the making of protein chains that happens in the ribosome, commonly associated with tRNA, rRNA, and mRNA. Codons that are copied from the DNA enter the ribosome, and the tRNA form protein chains from the codons.

Gene expression: the copying and editing of mRNA. mRNA copies one strand of DNA, edits it (cut out the introns that are simply 'junk, repetitive DNA' and paste together the remaining exons, or the genes that are 'expressed'). The genes are grouped into three-base sets, called codons.

After an mRNA is edited and its 'gene expression' is regulated, it exits the nucleus of a cell and enters the ribosome, where 'protein is synthesized according to the codons of mRNA.'

Protein synthesis is one of the possible outcomes of expression of genes that code for protein segments.  However, DNA also codes for RNA that is not involved protein synthesis directly.  These pieces of RNA act to regulate the expression of genes.