Both process maturity assessments and process capability assessments measure process performance and effectiveness. Both terms are used interchangeably but refer to the same thing.
Historically ITIL assessments have adopted the SEI (Software Engineering Institute) CMMI assessment approach, which combines capability and maturity assessments into a single assessment. Most consultancy firms still call it a process maturity assessment because the property assessment methodologies of these consultancy firms are based on the SEI (Software Engineering Institute) CMMI approach.
ISO 15504 refers to this as process capability instead of maturity. The ISO15504 approach is a more objective, repeatable, consistent, and robust approach to process assessment.