Difference between Data Science and Data Mining is going to be very interesting.
There are not so many differences between them, I should say the similarities between them.
Talking about Data Science, I have already discussed What Data Science actually is. If you have not check yet my previous posts, Please go and Check them out, you are missing something. Data Science is an umbrella that contains many other fields like Machine learning, Data Mining, Big Data, statistics, Data visualization, Data analytics etc. Below I have listed those posts, Visit them –
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Now, Let’s talk about What is Data Mining?
Data Mining is the process of gathering information stored in the database that was previously obscure and unknown. This information can then be used to make appropriate business decisions. Machine Learning is the technique that is broadly used in Data Mining. The essential objective of the data mining process is to extricate data from different arrangements of information trying to change it inappropriate and justifiable structures for inevitable utilize.
You look for consistent patterns and / or relationships between variables. Once you find these insights, you validate the findings by applying the detected patterns to new subsets of data. The ultimate goal of data mining is prediction.
Main Differences between Data Science and Data Mining –

Data Mining is an activity which is a part of a broader Knowledge Discovery in Databases (KDD) Process while Data Science is a field of study just like Applied Mathematics or Computer Science.

Often Data Science is looked upon in a broad sense while Data Mining is considered niche.

Some activities under Data Mining such as statistical analysis, writing data flows and pattern recognition can intersect with Data Science. Hence, Data Mining becomes a subset of Data Science.

Machine Learning in Data Mining is used more in pattern recognition while in Data Science it has a more general use
Data Science Vs Data Mining Comparision Table
Data Science 
Data Mining 
It is an area.  It is a technique. 
The main focus is on the Scientific study.  The main focus is a Business process. 
Its goal is to build Datacentric products for an organization.  Its goal is to make data more usable. 
A person needs to understand Machine Learning, Programming, infographic techniques and have the domain knowledge to become a data scientist.  Someone with a knowledge of navigating across data and statistical understanding can conduct data mining. 
Data Science consists of Data Visualizations, Computational Social Sciences, Statistics, Data Mining, Natural Language Processing, etc.  Data mining can be a subset of Data Science as Mining activities are part of Data Science pipeline. 
It deals with all forms of data – structured, semistructured and unstructured.  It deals with mostly structured. 