The difference between triggers and assertions are:
Triggers - a trigger is a piece of
SQL to execute either before or after an update, insert, or delete in a database.
An example of a trigger in plain English might be something like: before updating a customer record, save a copy of the current record. Which would look something like:
CREATE TRIGGER triggerName
INSERT INTO CustomerLog (blah, blah, blah)
SELECT blah, blah, blah FROM deleted
Assertions - An assertion is a piece of SQL which makes sure a condition is satisfied or it stops action being taken on a database object. It could mean locking out the whole table or even the whole database.
Triggers - Triggers are more powerful because the can check conditions and also modify the data
Assertions - Assertions do not modify the data, they only check certain conditions.
Triggers - Triggers are linked to specific tables and specific events.
Assertions - Assertions are not linked to specific tables in the database and not linked to specific events.