Integrated amplification means that the circuit inside is based on op amps and/or all-in-one modules. I refer to complete power amps inside one single plastic cased module, such as say Sanyo's STK xxx series.
Discrete amplification means the circuit uses individual transistors to reach the desired effect. Such a circuit is by default physically larger than an integrated module, which is very compact.
Integrated modules are the way to go if you want power on the cheap. All they need is a heat sink, typically several discrete components (usually a few resistors and capacitors) and a power supply. Since you'll need the heat sink and the power supply anyway no matter how you make it, it's obvious integrated amplification is a quick way to go about it. Also far cheaper, since your manufacturing time is cut down almost nothing.
Another benefit integrated modules offer is reasonable matching, especially in terms of heating up; since everything is on the same substrate, it's matched fairly well by default, and its thermal drift will also keep it in balance by default.
On the down side, they are shut in, no tweaking, no changes, they are what they are, period. Your control over their sound is reduced to power supply design and that's it. Their most serious shortcoming are their output stages; you can put just so much power on just one substrate. When they appeared, in mid-70ies, they were horrible sounding, but over the years, they have steadily improved to the point where some of them are known to produce quite reasonable sound. Reasonable, but not great.
Discrete circuits are much more of a hassle, because you have to design each stage on its own, and then make it work in the circuit as a whole. They will take more space, they will take more time to develop and more time to manufacture, consequently, they will be more expensive. And they require you to match some devices, primarily input and output stages.
Their good sides, on the other hand, are many. You can exercise full control over their sound one stage by another, you can use different transistors while tweaking to get even better sound, you can optimize each stage for its intended function and you can make its output scalable. As a simple example, if your supply is +/-35V, using an integrated output will get you your 50W/8 ohms, but what happens when your load impedance drops? As opposed to that, making a discrete circuit allows you to use not one, but two pairs of high power, high linearity output devices, which will indeed drive 4 ohm loads just as well as 8 ohm loads, delivering the required 50/100W into 8/4 ohms.
Whether you can hear the difference depends on many things, but by and large, you can. Typically (but not necessarily), discrete amps will deliver fuller, deeper bass with more authority, the mid will be typically about the same, while the treble will again probably be better on a discrete amp. The reason is speed - integrated amplification is not known for its speed, yet a discrete circuit can be made to work extremely quickly. Consequently, its phase shifts in the audible range will tend to be smaller to inaudiable.
Please bear in mind that these are genera comments only. Simply going discrete is no guarantee any amp will sound better by default, audio history is full of discrete amps sounding bland and uninvolving, but on the other hand, off hand, I'd say at least 99.9% of amps ever considred great were discrete and not integrated designs.
Cheers,
DVV