They are both fairly standard comments to make to people, both in conversation and in writing, along the lines of "how are you?" They are rather impersonal, as the "how are you?" question often is, and are non-committal, or do not require a response from the other person. The word "fine" is as plain and impersonal as it gets. Depending on the relationship between the people and the context in which it's said, "fine" can be totally acceptable, merely polite, or cold, disinterested and rude. If you are looking for a casual pleasantry to use in conversation or emails, I would go with "hope you're doing well" to avoid using the word "fine."

Both questions express your good wishes for the other person's well being so there isn't much difference between the two. They are just two ways of asking the same question.