Answer: This is a fascinating question which has occupied the audiological community for some time now, because in theory both techniques arise from the same neural generators but in practice there have been discrepancies in test performance (speed, efficiency and accuracy of threshold estimation).
The reasons for the discrepancies in the past are complex but a large part of the explanation centres on whether one or both techniques have used optimal stimulus and recording parameters. If this is not always the case, this could explain performance differences on direct comparison. Other reasons might relate to historical familiarity with the ABR, including the training materials available for audiologists and their equipment (e.g.with evoked potential platforms that have traditionally been centred on ABR testing in terms of software layout and functionality, for example).
On the other hand, ASSR offers clear advantages in terms of potential time savings by presenting test stimuli simultaneously, and by providing a fully objective outcome which is not exposed to errors of interpretation in the same way that ABR can be.
What is reassuring from the clinical perspective is that these two separate but related evoked potentials can now be used either interchangeably or in a mutually supportive manner according to the needs of the patient or other circumstances, giving the audiologist great flexibility in decision making.
Addendum, April 2018
A recently published multicentre trial (Sininger et al., 2018). has compared the 90 Hz ASSR and ABR in the clinical setting using the same CE-Chirp® stimuli. This is a crucial point as, although the stimuli rate is different for a transient versus a steady state response, the other features of the stimuli are the same between the two approaches. Other features of the recording and clinical protocol are described in detail in the paper. The results showed that average test time was around 20 minutes for the ASSR (for thresholds predicted at four test frequencies in each ear) but just over 30 minutes for the ABR to provide the same information. As mentioned in the earlier answer, it is also the case that the ASSR is fully objective whereas the ABR retains a degree of subjectivity related to tester interpretation of waveform morphology (and therefore potentially tester bias), although tools for objective waveform interpretation are available in the Eclipse ABR test protocols (e.g. the Fmp statistic, automated residual noise calculation and waveform reproducibility (cross-correlation) analysis). All in all, at least based on test time for audiogram prediction, these results would tend to support use of ASSR. Part of the reason for the lower test time is that stimuli of different frequencies can be presented simultaneously in ASSR but sequentially in ABR.
On the other hand this should not be taken to mean that the ABR is in any way obsolete or out of favour. It seems likely given the weight of research and development, the extensive evidence base and highly developed clinical protocols and the provision of training and education in ABRs, they will continue to play a central role in audiological evaluations alongside the ASSR.
What is reassuring from the clinical perspective is that these two separate but related evoked potentials can now be used either interchangeably or in a mutually supportive manner according to the needs of the patient or other circumstances, giving the audiologist great flexibility in decision making.
References and caveats
Sininger, H., Hunter, L., Hayes, D., Roush, P., and Uhler, K. (2018) Evaluation of speed and accuracy of next-generation Auditory Steady State Response and Auditory Brainstem Response audiometry in children with normal hearing and hearing loss Ear and Hearing.- Publish Ahead of Print