Cochleagram vs STFT for CNN-based speech segregation
I was hoping someone would be able to give me an explanation
  hilighting some of the differences between cochleagrams and STFTs, and
  what their individual use-cases would be


STFT (short time fourier transform) is a technique that is used to obtain a time-frequency representation of audio signals. Spectrograms and phase vocoders are arguably the most important applications of the STFT.

The cochleagram, (which is really just a variant of the spectrogram), is also used to obtain a time-frequency representation of audio signals but has one major difference compared to STFTs (and spectrograms). 

Regular spectrograms have a constant channel (frequency bin) bandwidth, which is given simply by $f_s / N$ (the sampling rate divided by the number of samples ). In contrast to that, cochleagrams' channels have different bandwidths, which are chosen to mimic human hearing  (the cochlea). 
There're more than just one (empirical) equation to compute the cochleagram bands and one of them can be found here  ( https://en.wikipedia.org/wiki/Critical_band )

$$ERB(f) = 24.7 * (4.37 f / 1000 + 1) (eq 1)$$

In a cochleagram, the bandwidths grow increasingly larger with increasing frequency. Note also, that it's possible to create a cochleagram by using a normal STFT and then averaging the energy found in several frequency bins so that they match the bandwidth(s) given by eq #1. Alternatively, you can use gammatone filters, which are basically band-pass filters of different bandwidth.