The Snapdragon and Cortex A8 are both implementations of the ARM V7A architecture, but are completely separate designs. The Snapdragon is Qualcomm's version and is unique to Qualcomm, whereas the Cortex A8 is licensed by ARM to third parties.
It's the same as Intel and AMD both making x86 chips; the ISA is the same but the design is different. However, ARM is more than happy to let Qualcomm design thier own.
I thought Cortex A8 was TI's implementation of ARMv7 and Snapdragon Qualcomm's: the ISAs are the same but Snapdragon can run distinct cores at distinct speeds because of disjoint caches per core, beneficial for power savings, for example.
It's the same as Intel and AMD both making x86 chips; the ISA is the same but the design is different. However, ARM is more than happy to let Qualcomm design thier own.