Why is an analog circuit more susceptible to noise than a digital one?
It all depends on the application that you are talking about, but generally the reason is because a digital circuit is either on or off. So lets say that in order to measure a digital high, the minimum threshold may be around 2-2.7v for 5v logic. And to register a low, it may only need to be as low as 0.8v. So any small amount of noise is generally not going to affect those signals as much since there's a wide range where it can register a high or a low.
An analog signal is basically an infinite number of voltages that it can be; any small amount of noise introduced to the signal will change that voltage. With a digital signal you only care if it's high or low. With analog, you care about all of the voltages in between.
It gets more susceptible to noise the smaller the signal is. If you have a 2.5v analog signal and 1mV of noise, this may not be too big of a deal depending on the application. But if you have a 25mv Signal that you need to amplify it to 2.5V, you will end up with 100mV of noise on the 2.5v signal once it's amplified. That could be a problem.
This is why it's very important to have very little noise with the beginning stages of a small signal amplifier.