So, what are the differences?

Since it's on the hair, this black crud can be washed off, usually with just water in a minute or two. If there is
more fat in the food, a bit of isopropyl alcohol and/or soap will help quickly remove it. Because it is bits of food and moisture mixed in with saliva to create black crud, it may stain white hair.
Wendy has this because her tongue hangs out, soaking her chin in saliva. I use a flea comb to comb off the bigger pieces and then a cosmetic pad with saline for contacts to wash it. Then I use EyeEnvy liquid and powder on the area. If I do this ever 2-3 days, her chin will be spotless. (Unfortunately, I often skip it during the week because I have such limited time.)

Clearing up feline acne may take weeks to months. Vets will often 'express' the pimples, which one can do at home and is rather disgusting. But if the pimples are not intentionally popped, they will pop when you scratch the chin or when the cat rubs against objects. The open 'wounds' do bleed freely and will create a mess. (On a side note, the expression seems in most cases not to cause the cat much discomfort.) Once the infections are removed, the vet normally prescribes an antibacterial soap or medicine.
When James, Clancy, and Tolstoy developed this, I would express the pimples and then wash the chin with a dilute mixture of goldenseal extract. The goldenseal has a natural antibiotic property, but it also stains the hair yellow. (I could live with yellow chinned cats.) Once I figured out the source of the problem, it took about 3 weeks for it to clear up.
So why is there a confusion? Basically it comes down to careful observation and lack of distinction. Most owners and some vets see the black crud and immediately call it feline acne. They don't stop and determine it the problem is above or in the skin. Then they treat it with antibacterial soap and it clears up -- not because it was acne, but because the soap killed the bacteria on the hair.
To some, this is splitting hairs. (Sorry, I couldn't resist.) The problem is once the owner stops the using the antibacterial soap and the crud comes back, often the vet then uses antibiotics. I'm not a fan of antibiotics in general, and in this case, they usually won't work, so then the vet tries either stronger antibiotics or suggests steroids. It comes down to treating a cat with some serious drugs for a problem the cat really doesn't have!
So where does chin crud and feline acne come from? The most commonly sited explanation is plastic bowls. And this is true, but not true. The chemical nature of plastic itself has nothing to do with acne or chin crud. The problem with plastic is it scratches. Those tiny grooves allow bacteria to build up. So for many owners, switching from plastic to ceramic helps.
In my case, my cats already have ceramic plates and stainless steal bowls. With Wendy, the crud is a byproduct of her malformed jaw. While I can help keep it clean (and cleaning her teeth will also help by decreasing the bacteria in her saliva), I will not be able to prevent it.
With James, Tolstoy, and Clancy, it was the food. They got it in period where I was teaching full-time and going to college full-time, so I had little time to make food. I would often mix canned food with my raw food to extend it. Once I removed the canned food, the problem began to resolve itself.