Humans excel at understanding speech even in adverse conditions such as background noise. Speech processing may be aided by cortical activity in the delta and theta frequency bands that has been found to track the speech envelope. However, the rhythm of non-speech sounds is tracked by cortical activity as well. It therefore remains unclear which aspects of neural speech tracking represent the processing of acoustic features, related to the clarity of speech, and which aspects reflect higher-level linguistic processing related to speech comprehension. Here we disambiguate the roles of cortical tracking for speech clarity and comprehension through recording EEG responses to native and foreign language in different levels of background noise, for which clarity and comprehension vary independently. We then use a both a decoding and an encoding approach to relate clarity and comprehension to the neural responses. We find that cortical tracking in the theta frequency band is mainly correlated to clarity, while the delta band contributes most to speech comprehension. Moreover, we uncover an early neural component in the delta band that informs on comprehension and that may reflect a predictive mechanism for language processing. Our results disentangle the functional contributions of cortical speech tracking in the delta and theta bands to speech processing. They also show that both speech clarity and comprehension can be accurately decoded from relatively short segments of EEG recordings, which may have applications in future mind-controlled auditory prosthesis.