LibreOffice and OpenOffice come from the same stock but they're two different, self-contained, beasts so you can install them together on the same machine (along with Microsoft Office, if you want) and they shouldn't interfere with each other. Of course, you'll have to decide for yourself what you want to do about file-associations and quickstarters.
When this article was last updated The Document Foundation was using the 'fresh and still' distribution model. (It used be called 'fresh and stable', which I thought did what it says on the tin, but it seems there were concerns people might think that if 'fresh' wasn't stable then what else could it be but unstable; which it was not.) Anyway, 'fresh' is the young, beautiful version; it has the most up-to-date features but, although it isn't actually unstable, it is more likely to have a few cutting edge bugs. 'Still' is a couple of point versions behind 'fresh' and is like its pipe-and-slippers uncle; less bling and accessories but steadier and more dependable and less likely to do anything strange.
LibreOffice has a partial dependency on Java, which is a freely-available piece of software that is often installed on computers to provide underlying functionality to other software packages. A good share of software needs Java to work. LibreOffice system requirements specify that "For certain features of [LibreOffice] - but not most - Java is required. Java is notably required for Base". Base is the LibreOffice database application, comparable with Microsoft Access.
Java digs its self quite deeply into its host operating system and includes a sandbox environment for running untrusted code. This, naturally enough, presents the systems administrator with a number of security considerations that go way beyond those associated with LibreOffice its self.
The installation and maintenence of Java are beyond the scope of this article and are not discussed here. A word of advice for non-techies though: if Java is installed on your computer(s) make sure you keep it updated. It should pop up messages on screen whenever updates are available, and offer to install them itself. If you ignore these messages (or please-update-me messages from any other software, including Windows) and end up getting hacked then you know who to blame.
For the moment, LibreOffice has no auto-update facility and it doesn't support incremental updates. So when a new version comes out you have to download the installer manually, all 200-odd megabytes of it, and re-install whole lot - even if it is only for a minor point revision.
(Some Linux distros might update it automatically but that would be the package managers doing their thing and not LibreOffice its self.)
LibreOffice does have a basic online update feature but for the moment all it does is check from time to time to see if a more recent version is available, and pop up a message on screen to let the user know when there is. This is very useful fuctionality on a home- or one-man-band business computer but in a well set-up business network environment, where most users don't have administrative privileges, it becomes a pointless and annoying nag-screen.
Fortunately, it is possible to disable this feature or even remove it entirely and these options are discussed later in this article.
The Document Foundation's Deployment and Migration page states that you don't need to uninstall any old version of LibreOffice you have before installing the current one unless you're upgrading from a version before 3.4.5: "You do not have to uninstall previously-installed version of LibreOffice if it is version 3.4.5 or higher [sic]. Lower versions should be uninstalled prior to installation. If you do have an existing installation of LibreOffice, all your preferences will be preserved and that old installation will simply be overwritten."
Sounds simple enough, although a query posted in the questions section of the LibreOffice website got this reply: "If you plan to upgrade to new version inside the same 3.5.x, 3.6.x, 4.0.x or 4.1.x series then there is no need to uninstall LibreOffice before installing update. But if you would like to install latest version of LibreOffice, but you currently have some old LibreOffice version, then first uninstall current old version before installing new one." [sic]. This seems to imply that if you want to upgrade , say, from 4.0.1 to 4.1.1 you need to uninstall first.
This answer was provided by a person identified as L-user on 16th September 2013 and, over a year and a half later, has been neither modified, qualified nor contradicted. Now, L-user might well be a LibreOffice super-expert but I can't see any explicit statement that he or she is so I'll stay with The Document Foundation's own advice.