Noise Reduction:  With conventional analog aids, the circuits were not able to tell the difference between a noise and speech.  With digital aids, a digital speech processor is analysing the sounds coming in all the time.  Noise looks a certain way - typically staying the same (think of a fan noise or road noise).  Speech looks very different from noise.  It starts and stops, has sharp sounds and blurred sounds.  The digital processor can distinguish between the two pretty well and can reduce the volume of the noise without turning down the volume on the speech.  This gives the speech signal the advantage over the noise. 

Directional or Dual microphones: The best way to significantly improve hearing in noise is by utilizing more than one microphone.  Traditionally, hearing aids only had one microphone and it had to amplify sounds all around you (omni-directional).  With the addition of a second microphone on the hearing instrument, the first microphone could narrow it's scope and become directional - picking up sounds directly in front of it - while the second microphone can pick up sounds coming from the sides and the back, reducing their loudness.  Many digital hearing aids now have microphones that are adaptive, changing how they amplify based on the enivronment they “hear” around them.