Profusely is another way of saying "excessively." If you sweat profusely whenever your crush walks into a room, you might reconsider your ban on deodorant before asking him out.
Profuse is an adjective meaning, basically, "a lot." Add the -ly and there's the adverb profusely, used to describe something that's giving off a lot — of blood, sweat, and tears for example. You might bleed profusely after you cut your thumb and cry profusely when your sweetie doesn't even offer you a Band-Aid. Other words with similar meanings include abundantly, extravagantly, lavishly, and richly. The French author Francois de la Rochefoucauld said, “Nothing is given so profusely as advice.”