As an FI you have the privilege to train for a licence or rating. A CRI only has the privilege to train licence holders for an aircraft rating. Under the new EASA rules they will be able to train for additional ratings that they hold but so will the FI.
When JAR-FCL was first printed it stated that the FI was also a CRI(SE) but that disappeared at around AL1 or 2 nevertheless, the CRI has less privileges than the FI.
Perhaps the major difference is that an FI (R) must be supervised for all activities whereas a CRI, who cannot conduct ab-initio instruction, does not have to be supervised.