When you open a workspace with a newer Desktop version you will have the option to upgrade specific transformers when newer versions are available. This is done one transformer at a time in a very controlled manner where you can review the changes in the current and the new transformer. This is a nice feature, but can be labor intensive for large workspaces. You can choose not to do this, but then you won't take advantage of the better performance the newer transformers often offer.
While upgraded workspaces can be opened with older versions, upgraded transformers will be removed. Make sure you create a backup before you upgrade as Safe suggests.
If you have FME Server you want to be sure the Desktop version is the same as Server, or older. A newer Desktop than Server version will cause issues when run on Server.
New .0 versions are almost always faster. Usually new formats are added, so you want to check that if you are waiting for a new format to be supported. When a new .0 version is available, you can read about the new features and formats in the new features page and the blog. You can also check the detailed changelog, which can be found at the download page, but it is quite a lot to take in and probably better used to look for a specific topic.
All major releases are scheduled, with typically 3 a year - one every four months - we target our major releases around the months of March (x.0), July (x.1) and November (x.2).
This and more about releases can be found in the article FME Release Process and Versioning.
For workspaces that are used repeatedly in a critical production environment, I prefer to make sure that the upgraded workspace returns the same output as the not upgraded one. This can be done using FME, processing an test/develop dataset with both versions and compare the results. This can also be done using FME, the ChangeDetector transformer comes in handy. This usually returns no surprises, but if it does, you want to know about it.