They're very similar, but "desert" tends to imply that you had made some kind of commitment. For example, you can desert a family, desert from the army, etc., but if you drop a pencil and decide to keep walking without picking it up, then you have abandoned the pencil. You cannot desert the pencil, because you didn't make any promises to it : )

They are pretty much the same. Though, a suggestion would be that: To abandon - focuses on leaving something or someone . E.g. You abandoned your car in the forest. To desert - focuses on the fact that you are are leaving something that you probably shouldn't e.g. You deserted your family, or You deserted the army. Hope that helps :-)