In-context learning (ICL) has become a prominent paradigm to rapidly customize LLMs to new tasks without fine-tuning. However, despite the empirical evidence of its usefulness, we still do not truly understand how ICL works. In this paper, we compare the behavior of in-context learning with supervised classifiers trained on ICL demonstrations to investigate three research questions: (1) Do LLMs with ICL behave similarly to classifiers trained on the same examples? (2) If so, which classifiers are closer, those based on gradient descent (GD) or those based on k-nearest neighbors (kNN)? (3) When they do not behave similarly, what conditions are associated with differences in behavior? Using text classification as a use case, with six datasets and three LLMs, we observe that LLMs behave similarly to these classifiers when the relevance of demonstrations is high. On average, ICL is closer to kNN than logistic regression, giving empirical evidence that the attention mechanism behaves more similarly to kNN than GD. However, when demonstration relevance is low, LLMs perform better than these classifiers, likely because LLMs can back off to their parametric memory, a luxury these classifiers do not have.