Generally speaking, an integrated amp will have better quality and design of audio electronics than a receiver. And, it will have more versatility (more setup and switching options). That's why they're generally more expensive. "Integrated" means that they combine a preamplifier (or control amplifier, with all you switching and equalization) along with a power amp (the part that actually drives the speakers). Originally, these were only available as separate components, particularly in higher end audio systems.
A receiver is essentially an integrated amp with a built-in AM/FM tuner, with all of the basic functionality of a separate integrated amp (but probably with lesser quality/design audio electronics and switching versatility). You also don't have to buy a separate tuner to listen to the radio.
For most non-commercial applications, a decent receiver should be fine. It also takes up a lot less room than the separate integrated amp and tuner, as well as being less expensive.
The audio electronics in various top receiver brands (Onkyo, Pioneer, Yamaha, Denon, etc.) are pretty much equivalent to each other. But, Onkyo is known for engineering superior tuners. So, I would recommend you do pick up an Onkyo receiver. It should meet all of your audio needs.