The clue is in the name. Simply put, water is the difference.
Coined in 1800 by a french chemist, the word hydration from the Greek "hydor" meaning water, is the addition and ultimately absorption of water. An anhydrous product, by contrast, contains no water. This is a purely oil based product which is oilier to the touch.
Hydrous (water based) creams
Because oil and water don't mix naturally we use an emulsifying agent to connect the water to the oil component. Without this, you might end up with a tub of water with some oil floating on top! Often, an emulsifying wax will be used and agitating the oil and water parts together forms a stable cream, such as the Oh Goodness Hand & Body Cream.
Any product containing water will require preservation, as bacteria grow in water. Be wary of "natural" water containing products that do not contain a preservative, as these types of preparations can spoil and grow dangerous bacteria.
Anhyrous (oil based) creams
A rich, oil based product, such as the Oh Goodness Intense Moisturiser, is what is known as "anhyrous", or "without water". These types of products encompass both creams and liquid oil preparations. Salves and balms (including lip balms) fall into this category, as do many (but not all) body butters. They often contain a wax, or solid butter, and have a correspondingly oily or "greasy" feeling and take longer to absorb into the skin.
Anhydrous preparations are usually preservative free, as the lack of water means they won't grow funky bacteria like a hydrous cream, though sometimes an antioxidant such as Vitamin E is added to extend the life of the oils.
The difference between hydration and moisturisation
Beware marketing ploys that tell you that an oil based product can "hydrate". These claims make me a little crazy! By definition, they can't hydrate, as they contain no water.
They can, however, moisturise. The definition of moisturise is to "make something less dry", which can be achieved with oils. Moisturisation is all about helping your skin form a barrier preventing irritation and dryness.
Hydration, on the other hand, is the infusion of skin cells with water to improve skin elasticity. A good, hydrous moisturiser will be both hydrating and moisturising, whereas an oil based product can provide moisture, but can't hydrate.
So which is best?
Everyone will have a preference but my own personal opinion is that each of these types of products have their place. As I have aged I've definitely seen the need for a richer product, especially on my face.
I generally shower in the evening, and I use the Oh Goodness Intense moisturiser straight out of the shower on my face, when my skin is cleansed and still a little damp. That way I get all the benefits of the beautiful oils and butters to nourish, as well as locking in some of the water remaining on the skin. The absorption rate is slower than a hydrous moisturising cream, but the richness of the oil layer means I wake with soft, nourished skin.
For hands and body, I much prefer a fast absorbing, non greasy cream such as the Oat & Glycerine hand & body cream. I can combat dryness, moisturise and leave skin soft and supple without that longer absorption time. Plus the skin on my legs is prone to dryness so I need that extra hydration power.
Everyone is different
Bear in mind that everyone is unique. What works for me won't work for everyone. My daughter, for example, has very different skin to mine so I encourage her to experiment to find what works for her. And I would say the same to everyone. As with diet, exercise, and indeed all issues relating to personal care, experimentation and doing what is right for you will be key to your success.