Bi-amping is a mixed bag
Assuming we're discussing passive bi-amping here (meaning the speakers use passive crossovers, and each speaker input is fed a full-frequency signal), you may or may not hear a difference, depending on the design of your speakers.
In passive bi-amping you're feeding identical signals into two separate sections of the speaker, usually the woofer(s) section and the midrange-tweeter section. If you're running small, dual driver satellites (e.g. a 7" woofer and a tweeter) I can't believe bi-amping will make any difference at all, unless you have a very weak amplifier; perhaps a single-ended tube thing would qualify. You see, the efficiency of one section of the speaker is not necessarily 3db greater than the two combined. So adding another 3db of power (like you get from the second amp) may not buy you any additional headroom at all! Where does the power go? It heats the crossover network. The 40Hz power supplied to the tweeter in a passively bi-amped configuration is the same as that supplied to the woofer. And since the voltage is the same as you would use with single-amp configurations the net effect might be exactly the same with or without passive bi-amping. Well, you'll use double the electricity with bi-amping.
So why consider passive bi-amping at all? Well... some speakers are nasty loads on amplifiers, and the loads of the woofer and midrange-tweeter sections can differ a lot. Woofers with big voice coils and large magnet structure produce a signal called Back EMF in the circuit. Without the boring details, Back EMF is a factor in the effective load an amp "sees" in the speaker circuit, and makes the speaker more difficult to control. Also, modern multi-woofer bass sections often seem to be designed with rather low impedances for various reasons, and the load is often capacitive in nature. Low impedance capacitive loads are also difficult to drive, so, in effect, complex multi-woofer bass sections muck up amplifiers to some extent. Finally, generating 40Hz and deeper tones can use a lot of power. Many speakers that claim 40Hz response are actually only producing 86-88db per 2.83v at 1 meter into 4 ohms (typical), and the 40Hz bass that shakes the room often needs 105+db of level at 1 meter for the right effect. 17db of additional level requires over 100 watts into 4 ohms in the scenario I've just described, while the midrange-tweeter section may be snoozing along at only 85-90db, or 1-2 watts. So some people think the heavy load of bass signals can sully the fidelity of the mids and highs that need only a watt or two on average.
But what really happens? Well, a lot of this is hypothetical double-talk. With adequate amplifier power there's little or no measured evidence that passive-amping is better than buying one amp with double the power. Two 100watt amps will almost always cost more than one 200 watt amp of similar quality. The key factor is that for mids and highs the first 2-3 watts of power is absolutely critical, and a complex woofer section just might destablize an amp enough that someone could hear a difference with a dedicated amp for each section. But it's theory, and I've seen no measured evidence that the destablizations would be AUDIBLE. So if you have complex loudspeakers with difficult drive characteristics and you have money to spare you might hear a difference. Or not...
What do I think? I think that money is usually better spent on a higher quality single amp than two amps for bi-amping, unless you're already in the top eschelon of amp quality (think $5000+ for one or two channels). (And some people would say you just take the money and buy better speakers. But that's another discussion altogether.) And very few speakers are complex enough to even make a good bi-amping argument in the first place. Of course, if you have such complex speakers you probably have enough money to indulge a few eccentricities, like bi-amping.
