When finished working in CiviCRM, is there a practical difference in terms of the health of my database between simply closing my Civi browser window vs explicitly logging out? We run on drupal...
The server doesn't know that you closed your browser, so it considers that you are still looking at the last page; for a very long time.
The vast majority of users doesn't log out to explicitly inform the server they are out.Your drupal/webserver is going to discard stalled sessions when they stay inactive for too long, so it doesn't have an impact to logout or let the server "auto logout" after a while.
They are tiny differences if you close the browser while editing, as Civi might warn other users that you might still be editing to prevent conflicts (when two users edit the same content at the same time, you might overwrite each other modifications), so here, properly logout -or cancel your edit before closing- does help avoiding that other users wonder if you are still editing it or not.
There might be small security concerns to leave your session open, but assuming your civicrm installation is using httpS (and if not, you really should ssl encrypt your websites), it's not an issue either.
So in a nutshell: it's slightly better to logout, but as it means you will have to login more regularly, you might end up using less Civi -yeah, we are that lazy- I wouldn't advocate to logout, unless you aren't using your regular computer.
If you aren't the only one using the computer, you should always logout (so the next one using it won't be able to access Civi with your account).
To add to the answer above you could consider deleting browsing history when the browser is closed.
This would effect all sites however many businesses use this method, really depends on your use case and how far you want to go.