Because proteins are the "workers" of the cell, and because different cell types -- muscle cells, neurons, hepatocytes, lens cells, red blood cells, etc. -- are different types of cells **because** they have different sets of proteins in them.
When I say proteins are the "workers" of the cell, I mean that most everything that happens in a cell happens because of one or more proteins. Glycolysis is catalyzed by a set of 10 different enzymes, which are proteins. The Kreb's cycle requires protein enzymes, as does ATP synthesis in mitochondria. DNA replication requires proteins; transcription requires proteins; and so too does translation. Regulation of gene expression also involves proteins.
Proteins are not just enzymes; there are structural proteins, motility proteins, regulatory proteins, transport proteins, hormonal proteins, receptor proteins, defensive proteins, and storage proteins.