I don't use conicals very often, but one advantage is being able to get rid of the break material while keeping the yeast. You don't have to have a conical to do this, though. You can either chill while still in your kettle and rack off it or filter it out, or you can first rack into a carboy, let the trub settle out, then rack it into your fermenter. If you do the second approach, you should avoid aerating it until it's in the fermenter; I also blanket it with some CO2 for some extra insurance. But it only takes an hour or two. Mostly I try to do the first approach (recirc/chill while in the kettle, then run off directly to the fermenter). If you use a conical, you can do the second approach with just the conical. Let the trub settle, dump it, then start fermenting. Leaving some break behind is OK, it helps the yeast, but too much can give off flavors if you leave the beer on it too long.
I don't use irish moss much any more; I tend to use whirlfloc or breakbright. But same idea. Copper finings.