What @Clayton posted seems about right to me, for those terms, and for "data mining" being one tool of the data scientist. However, I haven't really used the term "data collection," and it doesn't strike me as synonymous with "data mining."
My own answer to your question: no, the terms aren't the same. Definitions may be loose in this field, but I haven't seen those terms used interchangeably. In my work, we sometimes use them to differentiate between goals, or methodologies. For us, data-science is more about testing a hypothesis, and typically the data have been collected just for that purpose. data-mining is more about sifting through existing data, looking for structure, and perhaps generating hypotheses. Data mining can start with a hypothesis, but it's often very weak or general, and can be difficult to resolve with confidence. (Dig long enough and you'll find something, though it may turn out to be pyrite.)
However, we also have used "data science" as a wider term, to include "data mining." We also talk about "data modeling," which for us is about finding a model for a system of interest, based on data as well as other knowledge and objectives. Sometimes that means trying to find the math that explains the real system, and sometimes it means finding a predictive model that is good enough for a purpose.