Squarewave - I want to clarify a few things. Hope it will help in you getting the system of your choice. I'm on the same journey as well.
There are no motherboards with GPU built into them. Nowadays, all CPUs come with integrated graphics. Please ignore the APU marketing speak - it is a fancy name for integrated graphics - i.e. the GPU and CPU are on the same die or chip.
Now, depending on the graphics horsepower you need, you can get an additional graphics card to act as your GPU (thus bypassing the integrated graphics in your CPU). You typically gain graphics performance but lose on cost, heat (hence, noise), and needing a larger cabinet.
GPUs typically need lots of memory bandwidth for good performance - which is why good quality discrete graphics cards typically come with high bandwidth GDDR5 memory. However, this is expensive.
Integrated graphics use a part of your RAM (DDR3 memory) as the graphics memory. Since this is typically lower bandwidth than GDDR5 (half the bandwidth, IIRC), you get lesser graphics performance. As ROC rightly pointed out.
There is a part-solution though. You can use fast frequency memory to make up for the reduced bandwidth. You can often find fast memory at fairly reasonable prices if you try and hit the "sweet spot" of price/performance (as is usually the case with computer components).
One more thing to note is that if you are going to buy a discrete graphics card with DDR3 RAM, it will be the same as running integrated graphics - as far as memory bandwidth is concerned. You might as well just go with integrated graphics with say, 8GB of reasonably fast RAM.
One more thing to note - and I have a different opinion from ROC here.
Integrated graphics have improved leaps and bounds in the last couple of years. This is for both AMD and Intel.
For an HTPC, one big advantage you get with Intel is that Intel's chips are manufactured in a significantly superior manufacturing process. Plus, they have a significantly more efficient design and a more highly integrated chip. What this means is that the big benefit of integrated graphics (in Intel's scenario) is that your system will run significantly cooler than anything from AMD. I'm referring to the new Haswell series of chips though - not the older generation. The integrated graphics would be called HD4600 or HD4400 (there are higher versions as well, namely HD5000, HD5100, HD5200 - although they are more exotic and expensive and difficult to find). You can read a
review of Haswell's HTPC capabilities.
The
concluding remarks in the review are worth reading, IMHO. And so is the
power consumption. The system draws less than 50 watts of power in almost all cases (unless you deliberate try maxing out CPU and GPU with a synthetic test - and even then it only consumes 80watts or so). That is quite phenomenal for a a system that is capable of doing full blown HTPC duty. Note - this is 50 watts power draw at the wall - including everything in your system.
What this will allow is for you to either build a really small mini itx system with minimal cooling needs, a low power picoPSU, and consequently very little noise. You could probably even run some of the low power variants with a laptop adapter.
Now lest I sound like an Intel sales person, AMD has some compelling integrated graphics options as well. Less expensive than Intel. And better graphics (typically 1.5x-2x for the same price, although that has come down recently). But with a significantly higher power draw. And a lot of their launches have been paper launches. They promised 45W power efficient Richland but they were never available in the market. Their Kaveri is long awaited (and Indian design!) - and in terms of graphics, should be the same graphics capability as a discrete 7750 (with DDR3 RAM). But that will only be their 100W model. In fact, I am waiting for Kaveri too.
The thing is - even 100W is still way lower than the power draw of a CPU + discrete GPU. Plus, it means less fans, more airflow, etc.
Sorry for rambling on. This is a complex field, and I don't understand a lot of it as well. Especially the HTPC software requirement bit. Just sharing some of my research I have done in building an HTPC for myself.