How does 2-mercaptoethanol lead to shift of the band to a higher molecular weight?
2-mercaptoethanol (2ME) reduces the disulphide bonds in proteins. If disulphide bonds are connecting two polypeptide chains ("intermolecular") then 2ME would cause them to separate and therefore instead of a higher molecular weight (MW) band you would get one or more lower MW bands. 

However, if there are disulphide bonds in the same polypeptide chain, they may cause the chain to fold up. This results in the polypeptide chain to effectively have a smaller hydrodynamic radius compared to the fully denatured chain. In such cases reducing the disulphide bonds using 2ME would cause the band to shift to a higher MW. 


  What is the easiest and lowest cost method to separate closely related proteins


This is a different question altogether and there are several techniques depending on how "dissimilar" or "similar" the closely related proteins are. 2D PAGE (with isoelectric focussing) is one technique that can offer higher resolution compared to a normal SDS-PAGE. There are many other chromatographic methods.