It's more of just a personal choice for what you prefer, and what your circumstances are.
Stay logged in
= Makes it quicker to access and post to the site. If you have others that have access to your computer, then it could pose a risk of them being logged in to the site as you.
= More secure to prevent others from being logged in the site as you. However, it'll take a bit longer to log in each time you visit the site.