What is the difference between Azure Purview and MDM \DQS in SQL server 2019?

MDS - master data services lets you manage datasets and add reference information to that data. For example your source system might have a list of 2 million stock items, but the items stored in the system are missing the weight or useful categorisation.

MDS lets you import these records and add additional data such as reporting categories and manage this in excel. Then you can use this for downstream reporting purposes. Or you can create completely new datasets that are required for reporting. AFAIK Purview doesn't do this at all, but there are lots of promises that it will become the platform for it, and it makes sense that it would be.

Microsoft SQL Server MDS uses rather old technology and I would not recommend it for new work.

Microsoft SQL Server DQS is even older and again I would not recommend it for new work. It seems like something they deployed to tick a box in Gartner ten years ago.

Purview's current functionality appears to be limited to data profiling and tagging. What tables and columns are in your databases? what's the data profile? OK now add some tags so end users can easily search and find these columns.

AFAIK Purview does not store any actual data (i.e. a list of stock items) except for limited row previews. It stores data about metadata. (what columns are in this table? how many rows does this table have?)

MDS enables sharing common information across different systems. It can be the source of truth or it can work as a hub, capturing inforrmation from one system and sharing to others. Also, it provides access control, versioning, workflows, and revisioning before publishing new content.

DQS provides cleasing, profiling in a easy way to manage your data that currently Purview is not capable at all (Azure Data Factory introduced a very limited funcionality also not funcional without doing a lot of coding).
There are 3rd party solutions in the market however they are extremely expensive while Purview is far from be able to replace them.

Purview, as of today, provides some assistance in data governance by storing and managing metadata, data lineage and access control but it stops there.

MDS and DQS are more important today than ever given the increasing amount of information with bad data quality that is collected daily by companies with issues with their data governance. They are excelent tools to integrate data contained in silos inside businesses while providing clean and uniform data across the board and reinforcing data ownership.