and also, remember that a piece of sandpaper or MM held with your finger is not a truely flat surface that is just taking the top of the ridges off as you sand. If you only sand with the lathe on, each grit is not only removing material from the high spots, it is also deepening the existing grooves by removing material from the inside of the scratch. This is what a quick lengthwise sanding eliminates (assuming that you do get rid of all the circumferential scratches).
I actually differ from Eric (sometime, consistency is not my strong suit) in that I usually move up a grit to do the lengthwise sanding. IE sand with 320 with the lathe on, then get a fresh piece of 400, sand lengthwise till all scratches are gone, then power on lathe and sand with the 400 till I get a nice consistent surface, turn off and get a new piece of 600 sand lengthwise, ect. Not sure why, really. Just seems logical that if you removed the 320 scratches with the 400 when the lathe was on, then the 600 will remove the lengthwise 400 scratches easily enough, no need to use the 400 again. Also, the next grit will be newer and sharper, and cut more easily for the hand sanding part, and for the sanding with the lathe on, that's not as critical.
I never really invested a lot of scull sweat into that part though, so I may be missing something obvious here. I may try it both ways next time just to see if there is any appreciable difference between the methods, but I tend to doubt it.