Each amino acid has a characteristic side chain, or R group that imparts chemical individuality to the molecule. These R side chains contain different structural features, such as aromatic rings, -OH groups, -NH3+ groups, COO- groups, sulfur containing residues etc. This variety in side chains causes difference in the properties of the individual amino acids and the proteins containing different combinations of them.
The polarity of the R group is another important feature and is one of the basis sets for classification of amino acids.
Furthermore, the nature of the R group dictates various physical properties of amino acids (and also of proteins they are made of), such as state, solubility, mechanical strength etc. by influencing the way bonding as well as folding of peptide chains take place during the formation of proteins.
Thus the nature of the R group influences structure and properties of amino acids and also, function of proteins in a big way.