Lanolin oil is lanolin that has had the heavier waxes removed. The resulting material is liquid at room temp. The difference between the two is similar to the difference between fractionated coconut oil (FCO) vs regular coconut oil.
Anhydrous lanolin is lanolin that has been heated to remove all water. Lanolin normally contains some water due to its chemical nature and to how it is purified from the raw wool fat.
I use the thick waxy lanolin in soap. I have always liked lanolin for my skin, so adding a small amount of it to soap seemed like a good idea, Too much, though and the soap feels too slick and slippery, so when I do use it, it's at a very small percentage.
I have never used the oil, nor have I ever seen it, and cannot guess how it compares.
I'm with Earlene -- I've only used regular lanolin in salves and soap. I don't know if the sap value is different. Nor do I know if there's any particular advantage to using lanolin oil except if you wanted a product that stayed liquid at room temp.
Ditto with Deanna and Earlene, I use the waxy lanolin in a few specialty recipes like shaving soaps. It gives a waxier feel to the puck and seems to leave a protective coating on the skin - cutting down on wind chapping/ sensitivity.
There's a small downside to its usage in that application though... my testers report than lanolin shaving soaps leave their brushes a little "stickier". DH now uses some regular soap on his badger brush to fully remove the residue after shaving, but feels the extra step is worth it.