How are low-noise transistors different from other?
These devices address burst, avalanche, flicker, and thermal noise.

Burst noise is the result of inconsistent ion deposition in the semiconductor fabrication process. It is reduced by escalating the selection/rejection criteria stringency, selling chips at different grades (eg: fast, slow); by changes in layout to better account for process variations; and by changes in the fab process itself to improve deposition homogeneity.

I think of Avalanche noise as amplified shot noise. Under reverse bias some electrons collide with the lattice in the PN junction depletion region with enough energy to form an electron-hole pair. Depending on the reverse bias voltage and junction characteristics, an avalanche breakdown may propagate, registering as a current spike. It is reduced by manufacturers by design and process changes to both increase the length of the depletion region (reduced field) and increase the energy required to free nearby electron-hole pairs.

Flicker noise, also called 1/f and pink noise, comes from "slow fluctuations of properties of ... materials"[1] during operation. As it is a sum of other sources of low frequency noise, it is addressed as these sources are identified. 

Thermal noise is directly proportional to temperature, so any change that lowers local temperatures improves this figure. For example, changing die package for better dissipation; or layout changes to spread out local current hotspots.