The simple way to explain the difference is that appropriations determines how much money can be spent, while authorization tells how to spend it. Except, actually, if any money at all is authorized for a given item (even a single dollar), then you can spend up to the complete appropriation. The analogy I usually use to explain this is to assume you have a child who is in college. You want the kid to study nuclear gymnastics, but your kid is rebellious and chooses to study underwater basketweaving. If you have paid for UB 101 thinking it would be just a minor diversion, well, you are likely to end up with a basketweaver and not a nuclear gymnast. But, as an appropriator, you could cut off all college funds and kiddo is out of luck.
So let's stick to appropriations. Every program has a program element associated with it, which is the line item that money is being requested for. There are detailed descriptions of exactly what those program elements are supposed to fund and where you find those depends on which appropriations bill is involved and what sort of program it is (e.g. research vs. procurement vs. sustainment). That's more detail than we need for this purpose, though it matters if you actually spend time reading these bills and don't want to go blind trying to find the info you are looking for.
The President's Budget Request will have numbers for every program element in it. Some of those could be zeros. But the subcommittees of the appropriations committees will alter those numbers as they create the bills. That is called the mark-up and it takes several weeks. When the marked up bill gets released, it will have reasons for the changes, such as removing money for being early to need or adding money because it will encourage American industry (which is kind of a code word for it having to do with something in some particular subcommittee member's district) or whatever. Much of the heavy lifting is done by their professional staff members, who are middle-aged, sleep-deprived, hard-working, and underappreciated. There are also various organizations and lobbyists who are trying to influence what the numbers should be. The key point is that the members of the subcommittees control what the actual appropriations bills look like. Hence, you can write to and call your congresscritters and other committee members to tell them what you think. And how do you find out who the right people are? It's all on congress.gov which has links to the various committees and lists their members.
The subcommittee bills go to the full House or Senate for approval. After both the House and Senate committees finish with their bills, they have a conference, which comes up with a compromise bill. They often just split the difference on numbers, but that isn't always the case. However, you can be reasonably sure that if either committee appropriated some money for a given program element, it won't get completely killed. Crippled, maybe, but not killed.
I was going to work through an example and even came up with three examples of programs to use (distillation of unicorn tears, synthesis of informaldehyde and businesscasualdehyde, and mining of unobtainium), but I realized it wouldn't add enough without getting tedious. And I certainly don't want anybody to think I'd ever be opposed to distilling unicorn tears.
Bottom line here is to contact your congresscritters, especially if they are on committees dealing with things you care about.