I will argue that doorway pages are a technique to get better search engine rankings, while cloaking is a technology that has many uses, one of which is go get better rankings.
The original doorway page technique was to create a page that would be attractive to search engines and cause good rankings, but that wouldn't be a page that users would find useful. As a result, the user would have to 'move through the doorway' by clicking on a link to get them to the actual content that the site owner wanted them to see. Early doorway pages were noticible by having a large type link at the top saying something like "Click Here For Information On Blue Widgets".
This wasn't cloaking, since the user could see the same content that the search engines could. From a webmaster's perspective though, it wasn't as effective, because the user had to take an extra step beyond clicking on the SERPS to get to the page the webmaster wanted the user to see.
One of those technologies would be cloaking, which you correctly identified as showing one set of content to the search engines, and another to users. Using cloaking, webmasters could remove the extra step of requiring the user to do something in order to see the target page.
On the other hand, cloaking can also be used in applications where doorway pages aren't involved. For example, Yahoo uses cloaking to show pure links to search engines in its directory, while showing tracking links to human visitors. In this case, there's no doorway page involved.
I hope this helps you understand the difference between the two.