Actually, most low to mid priced headphones claim to have incredible frequency response, but very few can reproduce the low low frequencies they claim. The main advantage you will notice with using good full-size headphones or high quality IEMs (in-ear monitors) is improved clarity in the midrange (voice) and treble extension (which gives a sense of "airiness"). Also, the soundstage will be dramatically better - sounds extend many feet around your head instead of inside your head, with clear positioning of the various stereo sound effects.
A very good mid-priced choice is the Sony V6 (about $65). Shure and Ultimate ears make superb IEMs (I own the Shure e4). The best inexpensive buy are the Koss KSC35 ($30). I use them all the time despite having three pairs of $$$ phones. They are light, very comfortable, and sound amazing for the price. Great bass too - music sounds excellent.
