D'une part, ... D'autre part... = "On the one hand..., on the other hand...", while D'un côté... De l'autre côté... = "On the one side..., on the other side..."
Explanation:
While "D'une part, ... D'autre part..." is generally used when debating the merits of something, whereas "D'un côté... De l'autre côté..." is more typically used for expressing two different perspectives.
