Instead of having a composite primary key (this table maintains the relationship between the two tables which represents two entities [two tables]), the design is proposed to ha
There are lots of tables where you may want to have an identity column as a primary key. However, in the case of a M:M relationship table you describe, best practice is NOT to use a new identity column for the primary key.
RThomas's link in his comment provides the excellent reasons why the best practice is to NOT add an identity column. Here's that link.
The cons will outweigh the pros in pretty much every case, but since you asked for pros and cons I put a couple of unlikely pros in as well.
Cons
Adds complexity
Can lead to duplicate relationships unless you enforce uniqueness on the relationship (which a primary key would do by default).
Likely slower: db must maintain two indexes rather than one.
Pros
All the pros are pretty sketchy
If you had a situation where you needed to use the primary key of the relationship table as a join to a separate table (e.g. an audit table?) the join would likely be faster. (As noted though--adding and removing records will likely be slower. Further, if your relationship table is a relationship between tables that themselves use unique IDs, the speed increase from using one identity column in the join vs two will be minimal.)
The application, for simplicity, may assume that every table it works with has a unique ID as its primary key. (That's poor design in the app but you may not have control over it.) You could imagine a scenario where it is better to introduce some extra complexity in the DB than the extra complexity into such an app.