Data science, data mining, machine learning, all of them are very close fields and probably most people won’t know the difference and for all intents and purposes there isn’t any.
But data science is more of a field that studies ways to use data to your advantage. For example, I have millions of user generated log files from a software. If I study them I can probably find out what people do the most and maybe use that information to make a feature that makes the software better. Now how you study them is a different question that data science tries to answer.
Machine learning is a set of learning techniques used to essentially figure out function F in F (X) = Y (some cases Y is not available and we move on to clustering but that’s slightly off topic).
Data science uses machine learning or machine learning is applied to data science but you can see there is a distinction here. Data science can include along with machine learning, other methods, hypothesis testing, model fitting, eye balling the answer, etc.