The easist way to think about this is in the context of milk.

Milk is comprised of about 80% casein protein, and 20% whey. Normally, you couldn't visually differentiate the two: it just looks ... well ... liquid and "milky".

However, when you add an acid (i.e. vinegar) or enzyme (i.e. rennet; used in cheese production), the milk will curdle. This breaks apart the whey from the casein proteins.

The result is that the whey (a semi-clear liquid) will rise to the top, while the casein will usually sink to the bottom (it is heavier and coagulates in the presence of an acid or enzyme).

In cheesemaking, the whey is siphoned off from the casein, and the solids are then drained through cheesecloth, further extracting any residual whey or water and leaving the firm casein solids, which are then shaped into blocks of cheese (the rest of the cheesemaking steps are omitted here-interesting but irrelevant).

Cottage Cheese: Whole Food Example of Casein and Whey

For a whole food, real world example, just think about cottage cheese (Little Miss Muffet was quite fond of her "curds and whey").

Cottage cheese is basically milk that has been separated into casein and whey via enzymes.  The curds (solids) are the casein and the liquid part is the whey. For this reason, when you eat cottage cheese you are getting both whey and casein protein.

Here's the kicker: Outside of cottage cheese, for years whey was typically discarded or fed to livestock.

One day scientists realized that the whey contained a very high level of fast digesting, bioavailable protein. There was a rush to figure out ways to turn the liquid into a powder, and the entire whey protein craze took off.

Difference in Whey Versus Casein

Whey and casein have very different chemical properties.

While whey digests quickly, casein can take hours to be broken down and absorbed. Casein is poorly dissolved in water and cannot be denatured (meaning it won't change it's natural structure with the application of heat or chemicals.)

This makes whey an ideal protein right after you workout, when you want to make sure plenty of protein is available to assist with recovery.

Casein is also poorly digested by the human body, which is a contributing factor to why dairy is so unfriendly to the human intestinal tract.

When Doesn't Whey Have Casein?

If you consume whole food sources of dairy,  you'll be consuming both whey and casein together.

However, if you take a 100% whey protein powder, there will be little to no casein present in the protein, aiding in the digestibility.

A 100% casein protein powder, will have isolated any residual whey out of the casein. The reason for this is that the filtration and isolation process more or less removes all traces of fat and the other protein.
Casein molecules are "larger" and hence won't make it through most whey filtration processes.

Milk protein isolate products may also contain some whey, as well as protein powder blends that intentionally contain both casein and whey protein. These are both exceptions to that standard.

Protein powders that are labeled  "Whey Protein Concentrate" may also have trace amounts of casein, since the filtration process used to make concentrate products doesn't capture all of the casein.

However, if you are consuming whey isolates, there will be very little cross-over between the two.