Where we will go for ADB adapter and JDBC Activities?

ADB adapter is a technology adapter, which is is used to trigger a process at some event.And this sends message to ESB at Queue or Topic when an insert or update occur on the given table.And a message is sent to the Queue or Topic.
Whereas, JDBC palletes are simple pallete whcich are used fo JDBC Queries and and JDBC Insert statments or Stored Procedures in the database.

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There is a huge difference between ADB adapter and JDBC palette.
JDBC palette activities are used during process design in designer in order to perform various operation related to database like basic insert , update , select queries etc.
whereas the ADB adapter a tool that can be used for event based communication between database and the ESB. If the adapter has been configured for some database event (i.e insertion of a row in some table) , then the adapter will notify ESB about this event and vice versa ..

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There is a lot a functional advantages and disadvantages between those two.
If you have the data load some where between low to medium, then u should go for JDBC paletttes otherwise go for ADB adapter, basically it is well designed to handle medium to high data loads.
if you use ADB adapter, the data load will be on the adapter archive instance running on the admin not the process archive..
more over u cannot implement roll-back mechanism just like JDBC transaction group, whereas JDBC palettes handles this scenario much better way..