1. Minimize: This option only minimizes the web part for that user in the same session. If the same user comes again to the same page again he will see the web part.
2. Delete: This option will delete the web part from the page and no body will be able to use this web part. The web part will not be loaded into the page.
3. Close: This option will close the web part for everybody. But with this option the web part is still loaded on the page but it is just not displayed. For this web part resources are allocated that are never used. We can also close the web parts by clicking on 'X' button.
We can see the closed web parts by appending ?Contents=1 at the end of the URL of the page.
But I struggled to restore these closed web part. Surprisingly the reset option which comes on the web part maintenance page also does not do this. Then after some trials the following steps worked:
1. Go to Site Actions and then to Edit Page
2. Now Click Add a Web Part link
3. Go to Advanced Web Part gallery and options
4. On top of this pane you will see Closed Web Parts. Once you click this it will show all the web part which are closed on this page.
5. Now Drag closed web part onto the page.
These steps will put the closed web part back on the page.
Attempt a Question on WebParts: Question on Web Parts