The processors themselves are just as stable. You can make a stable or unstable system, whether it is based on an Intel or P4 chip, it all depends on buying the correct, high quality power supply, cpu cooler, motherboard and memory. There is a reason you might want to go with Intel, but it is not neccesarily stability; the chipsets used on modern motherboards for AMD (and some of the cheaper Intel motherboards as well) are made by VIA, and although they are capable nowdays of making stable chipsets, they do have some issues regarding throughput of the PCI bus. This could be important if you plan on using several drive simultaneously.
I place a lot of importance on quiet PCs, and therefore use a Zalman 6000Cu heatsink and 120mm fan on my XP1800+ cpu, along with a Seagate Barracuda IV hard drive. I have also modified my Enermax power supply with a quieter fan and lined the case with insulation. The system is very quiet and at night when there is almost no noise, I can just about hear it from my bed which is around 10 feet away.
Check out the hardware at www.quietpc.com for silent power supplies and CPU fans.