What is the difference between Acrobat SDK and the JavaScript API?

An SDK is a collection of info for programmers. So the Actobat SDK is the info all programmers wanting to work with Actobat will need. It's mostly documentation, thousands of pages to read.

Part of the SDK has the information Acrobat JavaScript programmers will need.

You won't be working in a browser. JavaScript is used to add on programming extras to different apps. So it adds on to browsers. And it adds on to Acrobat. But the two are quite separate and different.

There is is no phone support for programmers. Experienced programmers who are stuck might buy $200 support cases, but Adobe won't teach programming.

Your task ask can be done in a number of ways but not purely JavaScript as it can't write databases. Do you want a server solution for submitted forms or something to use on forms you have on your computer? (Note: don't make assumptions about someone else getting a form to you)

There are different parts to the Acrobat SDK: The most important ones are the plug-in API to write Acrobat plug-ins in C/C++, then there is the IAC (or InterApplication Communications) API, and the JavaScript API.

The SDK contains mostly documentation and some sample code, so the JavaScript API is a part of the SDK.

No, you are not using JavaScript in the browser, it's running inside Adobe Acrobat (or the free Adobe Reader, but there are limitations in what you can do with JavaScript within the Reader). Take a look here: Acrobat DC SDK Documentation | Adobe Developer Connection

You will see a section titled "JavaScript for Acrobat". The first document listed is "Developing Acrobat Applications using JavaScript" - read this to get more familiar with what you can do with JavaScript in Acrobat.

There is no phone support for this. If you run into a problem, you can raise a developer support case (and pay for that), but chances are that you get help faster and better by just posting in the JavaScript forum: JavaScript

To learn JavaScript, you can use any resource that teaches JavaScript for the browser, you just need to know when to put that resource aside because it is getting browser specific: Every implementation of JavaScript consists of two parts: The core language and application specific extensions to the core language. The core language is the same for all implementations (that is, as long as you ignore differences due to engine version changes). The application specific part is where you can ignore the browser specific parts of JavaScript, and then use the JavaScript for Acrobat API Reference (linked on the SDK page).

If you are looking for an Acrobat specific introduction to JavaScript, take a look here: Beginning JavaScript for Adobe Acrobat  - this is not a free resource however.