Some applications detect old versions of themself and install over them, so I'd install the new version first and see what happens. If you end up with both versions of Lightroom. If you do, then delete the old one.
The majority of the application should be gone when you delete the app form 'Applications'. There will often be a small number of small files (usually configurations and user preferences) left over. These are often left behind so that (1) if you re-install the app later it will have your old saved settings (2) if you were using a trial version that expires it knows when it was originally installed.
Personally, I would find and delete those files myself. I know I'm paranoid, but I have little faith that 3rd party clean up tools will get rid of all of the old "crud" AND won't accidentally delete something I want/need.