Just a preliminary response, but you can start with the fact that Business Intelligence does not specifically mention Big Data, while your other term does. So one difference, clearly, is that Applied Big Data Analytics concerns Big Data, and BI does not (necessarily). BI is a general term that encompasses a stack of technologies and the techniques used for making better (data driven) business decisions. Applied Big Data Analytics is not an "industry standard" term, as such. So I see it and I think it is the application of analytics to big data, but this could be for any purpose, not necessarily for business decision support.
Both the processing and analysis of data collected in Big Data database systems and analytics conducted on computerized Business Intelligence platforms can support the company's management processes. However, not to the same extent and not in the same applications and fields of management. The difference is the processing and analysis of data collected in Big Data database systems can have very different functionality which is determined by the type and data sources stored in Big Data database systems. However, the analytics conducted on computerized Business Intelligence platforms is focused on specific aspects of the business, economic or financial and financial analysis of the company.
The issues of the use of information contained in Big Data database systems for the purposes of conducting Business Intelligence analyzes are described in the publications:
It heavily depends on your point of view. From a technical perspective, techniques used in classic BI and the techniques used in Big Data Analytics are totally different. So, from that perspective, you should never say BI=Big Data Analytics just because you are using large amounts of data.
On the other side, from the user perspective (e.g. the manager looking at the latest reports or findings), one cannot tell whether the provided information has been computed with traditional techniques (e.g. ETL, DBMS, Data Warehouse, OLAP, etc.) or with big data technologies (e.g. ELT, NoSQL, Spark).
For the purpose of BI, one can even mix traditional techniques with big data techniques: you might come up with new BI Front End tools (e.g. web-based applications) to directly access the information computed using big data technologies (e.g. using Spark to process big data and using NoSQL databases to store the analysis results in the Batch and Streaming Views ). Or big data experts might extract relevant information out of the NoSQLs and store it in the existing BI technology stack. That way BI domain experts responsible for generating new reports or adjusting existing BI analysis queries don't have to deal with technologies which they are no familiar with.
So, if the amount of data becomes very large and traditional BI techniques start to fail, big data technologies are required. Still the user of BI (the managers) doesn't see the huge complexity behind the scenes...
So, can I say, BI is the traditional tool that has been integrated in the new tool Big Data Analytics ? BDA use BI platforms in addition to advanced tools ?
I don't think the two terms are completely interchangeable, the way that you seem to want to use them. Personally, I have not seen or heard "Applied Big Data Analytics" given a similar industry standard usage, definition or general acceptance as Business Intelligence. I also don't think you can assume that one has been absorbed or integrated into the other.
BI might be concerned with Big Data, or it might not. It will depend on each individual implementation. Nothing says it can't be, nothing says it must be. It almost always refers to a stack of technologies that collect data from transactional data sources, pools them into query optimised storage and then presents views of that data that are useful to a non-technical user.
ABDA clearly is concerned with Big Data, evidently from the phrase itself. Analytics is a general word for various techniques (descriptive stats, clustering, pattern recognition). So just by looking at the phrase itself, you can see it means the application of these techniques to Big Data. The reason why that is a specific application or research area, is that Big Data requires special techniques to overcome the specific challenges of Big Data itself - often referred to as the five Vs of Big Data (Volume, Velocity, Variety, Veracity, Valence). There are more challenges than these, but these are considered the classic problems to solve. So ABDA is concerned with overcoming these challenges to apply analytics to Big Data.
As such, I think the two phrases you are looking at are quite separate and distinct. You might say that BDA should be used in a BI system when that system includes Big Data sources.