First of all, let's define the term 'science' in data science.

In general, science can be broadly defined as a logical system of getting and organizing insights in some area of knowledge as a set of some principles, explanations that can be used for making some previously unknown but testable predictions. In order to qualify as a principle, an insight must be both highly general (applicable to many settings) and stable (relevant now and in the future developments).

From this standpoint, statistics is a science (statistical science). It is an integral part of applied mathematics with its own principles such as, for example, Central Limit theorem, principles and methods of unbiased sampling, etc.

What are the specific data science principles that make it a real science? Not too many.

Data science in its current state is rather a loose collection of various computational methods applied to big empirical data sets collected without pre-planning (just because a lot of data are available), such as supervised, non-supervised, neural networks algorithms, etc.

The claimed purpose of data 'science' is inferring some insights (quantitative and qualitative) from data. However, data 'science' does not offer a causal explanation of any relationships between the variables (factors, features) but it explores merely some empirical patterns, associations / correlation of the past data and attempts making a formal projection of these data to forecast some future values.

On the other hand, is the future forecast always possible based solely on the past? No, in general prediction or forecasting the future solely from the past is not possible regardless of the amount of the past data and regardless on the individual’s role. It is possible only in a limited number of cases in which the stable past data pattern can be reliably extended in the relatively short future horizon.

Data analytics is much more defined and technical area.

What is data analysis? Any use of the word 'analysis' requires a clear statement of the questions that you want to get answered.

Only if you have a right question you could analyze your data, and your analysis means that you could get an answer to your questions using some mathematical methods or computer simulation. However, you will not get the meaningful answer if your question is wrong, or your data do not contain the right information for answering your question. This situation is typical if data are collected just because they are available without a pre-planned data collection procedure.

Next, getting simply a descriptive statistical summary such as data mean and standard deviation is NOT the same as data analysis. It's just descriptive statistical data summary.

There are two distinct areas concerned with using the data analytics potential. One is focused on Technology for storing, processing and managing large amounts of data of various nature in some data base form. This trend leads to fitting a company’s arsenal with data-savvy tools. Value is too often considered as something that increases solely by the collection of more various data. This results in investments in data-focused activities around the tools. This leaves an organization with a big set of tools, and a small amount of knowledge on how to convert data into something useful for this organization.

Another area is Methodology for making business decisions using modeling and simulation based on data specifically collected to address some business problems.

The bottom line: data analysis always starts with the question you want to get answered, then identifying an appropriate method of analysis, i.e. the procedure that will get you to answering your question, then collecting and preparing the needed data (beyond the simple descriptive statistical data summary), and only then feeding the data into the analytical procedure, and then validating results of your analysis.

Data Analytics focus steps: (i) Defining a business problem, (ii) identifying an analytic method (algorithm) or simulation approach, (iii) collecting data required to feed the algorithm, (iv) validating your solution (v) turning solution into the actionable managerial decisions.

The three (data science, data analysis and business intelligence) involve data. In fact, data science and business intelligence involve data analysis.

In the practice, there can easily be an overlapping of techniques as deemed necessary for a specific work.

Business intelligence is more focused on the output, visuals and end-user tools that a business consumes to benefit from frequently updated figures. It had its boom a decade or two ago, when the expertise was very expensive and demanded, although the market continues growing nowadays. It deals more with the structuring and standardization of data objects in a data warehouse, and sometimes cubes, with all its specialized source data processing (e.g. ETL, ELT) end to end methods.

Data science, on the other hand, has been there for much longer. It is more about the methodologies, scientific approaches and ad-hoc analyses than the tools. It normally does not spend too much on the look and feel of the output, but rather on the meaning and value of the insights or findings of experiments. It includes machine learning algorithms, models, data pipeline and exploration techniques for which you can find literature dating back to 50 years ago. There are data scientists and/or statisticians in laboratories or offices employing data science alike.

While it is acceptable that a data science assignment is done by a single person, a business intelligence work in a professional environment used to demand two or more experts (front- and back-end). However, the industry has been changing, and the empowerment of end-users with self-service is sacrificing process stability in exchange for faster, more agile results, leading to a combination of different practices. As recent example: super users or citizen developers using Power BI with Python for machine learning.

In case you meant data analytics (instead of data analysis), this deserves much more writing, but in summary: it covers a combination of multiple (if not all and more) of the above concepts. It deals with supervised and unsupervised machine learning, like in data science, but spanning from descriptive and predictive to prescriptive and cognitive data analysis techniques.

Each topic can have a full book of details and there will be multiple points to compare, with more similarities than differences in terms of data management practices.