This paper provides a systematic analysis and comparison of the performance of state-of-the-art models on the task of fine-grained Arabic dialect identification using the MADAR parallel corpus. We test approaches based on pre-trained transformer language models in addition to Naive Bayes models with a rich set of various features. Through a comprehensive data- and error analysis, we provide valuable insights into the strengths and weaknesses of both approaches. We discuss which dialects are more challenging to differentiate, and identify potential sources of errors. Our analysis reveals an important problem with identical sentences across dialect classes in the test set of the MADAR-26 corpus, which may confuse any classifier. We also show that none of the tested approaches captures the subtle distinctions between closely related dialects.