I'll make a very brief start, but the list could go on and on.

And, this is a really rough overview - the good books on this dedicate a page or two to every point I've mentioned.  And there is more than one type of communicative approach, so some people will have slightly different views about what it really encompasses.

Advantages:

• A focus on language in use in context
• A focus on meaning and purpose
• Encourages learning through doing (as opposed to 'rote memorisation' or 'gaining knowledge'), and therefore emphasises both real-world tasks and fun activities
• Sees errors as part of the natural developmental process
• It recognises that spoken language is different from written language and just as important (if not more important)
• It recognises that language is about interaction - there's no point speaking if no-one's listening, there's no point writing if no-one will read it, etc

Disadvantages:

• Most people get confused by its name and think 'communicative = spoken language'
• The 'pure' form* of the communicative approach didn't encourage focus on form or accuracy to any real degree; and research over the last 20 or so years has clearly demonstrated that some focus on form and/or accuracy is helpful. 

* Actually, the pure form was never really used anyway - teachers and course-book providers have always wanted to keep their grammar boxes, grammar exercises and repetition drills regardless of research)