To add some detail to what Jim said:
Update Tags (from Library) takes the internal MC Library meta data and writes it to the Tags inside your music files. Thus:
Update Tags (from Library): Library ---> Tags in Files
Update Library (from tags) does the opposite. It reads what is in the Tags in your Files and writes them to the Library. Thus:
Update Library (from tags): Tags in Files ---> Library
Why would you use them? Well, if you have made a bunch of changes in your MC Library (changing artists, albums, ratings, song names, etc) and you want the Files themselves to be updated, you would run Update Tags (from library) to push those changes to the files.
On the other hand, if you have used an external program to change the Tags in your Files and you want those changes to be recognized by MC, then you would run Update Library (from tags) to take the external file changes and push them into MC's Library.
Note that both of these can be done fairly automatically. Two settings control this:
1. Tools > Options > Library & Folders > Configure Auto Import > Tasks > Update for external changes . If this option is checked, it automatically picks up any Tag changes in Files that are in your Auto Import directories.
2. Tools > Options > General > Importing and Tagging > Update tags when file info changes . If this option is checked, any changes you make to MC's Library are pushed out to the File Tags automatically.
Note with all of the above that not all Library fields can be written to Tags. Anything custom or non-standard can't be written to tags because they don't exist. Certain file types support certain Tags. Only supported Tags can be written to.
I hope this helps.
Brian.