Data as an industry is growing so fast that old things have new meanings and new things have old meanings and things that seem new are in fact old. If you ask 10 people in 10 different companies in 10 different countries you’re likely to have many different answers. Different people with different views and agendas will give different answers.
Let’s start with data analytics or simply analytics since I can’t imagine it being done without data! Whatever you do to extract knowledge from data can be considered analytics, from the most simple statistical procedure to the most complex machine learning model, all of that, in my humble opinion are analytics if their purpose is to extract knowledge from data. Apart from that analytics is a vast generalisation.
Business analytics is doing exactly that to improve business performance using business data. This leads to specialisation. Many services and products exist to provide business analytics that is common to all types and sizes of business, for instance, ROI is common across all industries. However, each industry ends up having its own silo of metrics. As an example, mobile analytics grew from web analytics but now it is fair to say that apart from marketing that is somewhat similar in both, all user related metrics are very different with the explosion of mobile in general and especially mobile freemium games. So in a way business analytics is sort of applied analytics in the business context.
Last but not least, data science. I would say that the end product of data science can be categorised as analytics. What distinguishes data science from business analytics are the skills, specifically data skills both in the nature of the data and the output produced. By nature of the data, I mean that unlike other areas that rely on clean and “well behaved” relational data, e.g. business analytics, data science will deal with data of any size, structure and quality. Data scientists often become domain experts of their industry across many areas of the company. Last but not least, data science is extremely technical. Knowledge can range from programming to complex machine learning, mathematics and statistics. Learning new stuff all the time is a given.
Hope this helps but keep in mind that there are many different views on this.