Classification predicts a category, such as up or down; regression predicts a number, such as next week's return. Classification is generally more robust on financial data because it does not chase the magnitude of outliers, but it discards information about how big the move was.
The framing everybody underestimates: your training data has very low signal-to-noise. An accuracy of 52% on a balanced directional problem can be genuinely excellent and is easy to mistake for failure, while a reported 85% almost certainly indicates data-leakage.
Model choice matters much less than data quality, labelling, and validation design. A logistic regression on well-built features validated with purged-cross-validation beats a deep network on leaky features every time.
Related: unsupervised-learning, label, feature-engineering, cross-validation