What is the difference between experimental and amateur built aircraft in the US?
The FAA provides pretty clear definitions and/or explanations: 

Amateur built: 


  Title 14, Code of Federal Regulations (14 CFR), part 21, section
  21.191(g), defines an amateur-built aircraft as an aircraft "the major portion of which has been fabricated and assembled by person(s) who
  undertook the construction project solely for their own education or
  recreation."


They define Experimental as a category, since it has to do with how the airworthiness cert is issued: 


  A special airworthiness certificate in the experimental category is
  issued to operate an aircraft that does not have a type certificate or
  does not conform to its type certificate and is in a condition for
  safe operation.  Additionally, this certificate is issued to operate a
  primary category kit-built aircraft that was assembled without the
  supervision and quality control of the production certificate holder.


And further 


  Special airworthiness certificates may be issued in the experimental
  category for:   
  
  
  Operating amateur-built aircraft: to operate an
  amateur-built aircraft in which the major portion has been fabricated
  and assembled by persons for their own recreation or education.
  Operating kit-built aircraft:  to operate a primary category aircraft
  that was assembled by a person from a kit manufactured by the holder
  of a production certificate for that kit, without the supervision and
  quality control of the production certificate holder.
  


So... All amateur built aircraft are experimental (flown under an experimental airworthy type certificate) but not all experimental's are amateur built, for example a pre-production aircraft may need to be registered as an experimental for test flights before a type certificate is received.