Whats the difference between concrete and cement?

Portland cement is a gray powder that is the binding agent which, when mixed with "aggregate" - sand and lime (the mineral) for mortar, or sand and gravel for bigger jobs - makes up "concrete". Your job will best be handled with a concrete patch mix from the local home center. This stuff has cement, fine aggregate and resins to help it stick to the old concrete. Much easier, and it's available in small amounts. As for the dishwashing liquid, that sounds like a "home remedy snake oil" type of thing, unless he really had some binder resin that he kept in a squeeze-bottle for ease of use.