After the boil I have been told to run the wort through a filter before adding yeast and letting it ferment. It seems like you would get more flavor leaving the hop leftovers in.
What is the advantage of filtering the wort before fermentation?
It would remove the break material and other such things, which may or may not have an effect on how long the beer would store if you're planning on aging it. You may also filter out nutrients that the yeast could use if you're doing that, though. Filtering after fermentation makes much more sense to me.
As a matter of practicality, I'd be tempted to screen, not filter, and have in the past. I also use a wort-chiller of the counter-flow variety, and they do not work well with either the break material or floating bits of hops. A standard / fine stainless wire mesh strainer is plenty good enough to keep the chiller from getting clogged with solids of either sort. While the process of forcing wort through the clogged chiller in an effort to clear the jam can be entertaining, it's not recommended.