Steps to reproduce:
Burn a ATMEGA328P-PU with an ArduinoISP - No errors
Download blink sketch - No errors and it blinks
Download blink sketch a second time - Fails with following error:
avrdude: stk500_recv(): programmer is not responding
avrdude: stk500_recv(): programmer is not responding
Which bootloader code are you using. This sounds like on of the Uno bootloader bugs that surfaced a while back. There is a newer version around that solves it, search around. I don't own a Uno so I haven't kept a link.
I think it's because with a fresh bootloader there is no sketch to run so the bootloader just waits for your upload.
Once a sketch has been uploaded the bootloader only gets control for a short time after reset before turning control over to the sketch. If the upload doesn't start during that window the bootloader won't process it.