I used to asked me the same question every time I started a project.
Now the way I use them is if the site/web page can still function (or live) without certain functionality then I make that functionality (like a polling system) to throw Exceptions if there is something wrong with it. But if the site/web page is dependent on that functionality then I throw an error because it's more critical and I need to get to fix it quick.

I don't use my own error/exception handling. I just make sure I let the user now and let myself now (by sending an email, message, twit, ... whatever flows your boat).