What's the difference between protein and DNA behaviour during agarose gel electrophoresis?
There are two important properties that determine how a molecule will move during gel electrophoresis:


size and shape of the molecule
charge of the molecule


You're applying an electric field, and all molecules will move according to their charge. The gel itself hinders the movement of larger molecules. This means that small molecules will move further than large molecules. 

Nucleic acids like RNA and DNA have a uniform negative charge from the phosphate backbone. This means that they will generally separate according to their size and shape (plasmids can be present in different forms like supercoiled, circular or linearized). 

Proteins have different charges, some have a positive charge at the pH typically used for gel electrophoresis, some have a negative charge, and some have no charge at that pH. So if you would simply run proteins on a gel, some would move backwards, some forward and some not at all.

To make the proteins separate according to their size we use the detergent SDS to denature the proteins, and to give them a uniform charge (SDS is negatively charged). Proteins are also run on polyacrylamide gels and not agarose as the size of the pores in agarose is a bad fit for the typical sized of proteins.

While I've never seen anyone using agarose for proteins, it seems that it is possible and sometimes used for larger proteins. I found one paper describing a protocol, they used very high concentrations of agarose in the range of 4-5%. I suspect that they won't handle well, but you should try that out yourself. If the proteins you're examining are a bit larger, using agarose should be possible.