There is no difference. The "lithium" in lithium grease is actually from the reaction of lithium hydroxide or lithium carbonate with a triglyceride. Triglycerides are an ester of three fatty acids and glycerol and are obtained from vegetable fats or animal fats. The lithium base splits the ester into glycerin and a salt of the fatty acid chain. Your grandparents (or parents depending on how old you are) may have done this kind of chemistry to make soap. They may have used actual lye (sodium hydroxide) or just ashes from a wood fire. The result is a fairly harsh, hard soap.

Lithium grease takes this "soap" or surfactant and uses it to dissolve up oil in the solid to make grease. The mixture is likely 90% oil and the rest the surfactant and/or other stuff. The resultant mixture will remain stable for a very long time. Over time and exposure to the elements, however, the oil can start to bleed out. The surfactant is left behind. None lithium grease can be a mixture of surfactants or it can contain clay or both. When people open up old bearings and find "peanut butter" or "grape jelly" inside, what they are seeing is the binder (soap or clay) left behind. The oil didn't "harden", it flowed away over time.

Modern polyurethane greases use polyurethane as the binder. It still uses oil but the oil seems to hold in the polyurethane better than soap based greases to. It's more stable for longer.