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Let's say we have a binary classification task, but our dataset contains more fine grained values of how much an examples belongs to the class or not. So the labels are real numbers in $\left[0,1\right]$. I can see two ways to make use of this additional information:

Approach this as a classification problem and use the cross entropy loss, but just have non-binary labels. This would basically mean, we interpret the soft labels are a confidence in the label that the model might pick up during learning.

Frame this as a regression problem, where we want to predict the exact amount of how much an example belongs to the class. In this case, we would use a regression loss like MSE or Huber loss.

What is the difference between the two approaches? How do I decide between them?