What's the difference between feature importance from Random Forest and Pearson correlation coefficient
Pearson correlations capture linear relationships between the input and target variables. Therefore this only makes sense for continuous inputs and a continuous target variable, and not continuous inputs with a binary/categorical output. Correlations essentially measure the positive/negative 'change' in one feature as you increase/decrease the other.

So it doesn't make much sense to compare the relationship between your input features and the categorical outputs this way. You may as well calculate the mean input for each feature and each label, and calculate the differences between those. I found this answer on Cross-Validated which explains this much better than I can.

Feature importance in tree based models is more likely to actually identify which features are most influential when differentiating your classes, provided that the model performs well. How this feature importance is calculated depends on the implementation, this article gives a good overview of how different tree based models calculate importance for features.