Continuous model is same as staged model only that the arrangement is bit different. Continuous representation / model concentrate on the action or task to be completed within a process area. It focuses on maturing the organization ability to perform, control and improve the performance in that specific performance area.
Capability Level 0: Incomplete
This level means that any generic or specific practice of Capability level 1 is not performed.
Capability Level 1: Performed
Capability level 1 process is expected to perform all capability level 1 specific and generic practices for that process area. In this level performance may not be stable and probably does not meet objectives like quality, cost and schedule , but yes still the task can be done. It’s a first step, it’s like you are doing the process but you can not really prove it if it’s the most effective.
Capability Level 2: Managed
A managed process planned properly, performed, monitored, and controlled to achieve a given purpose. Because the process is managed we achieve other objectives, such as cost, schedule, and quality. Because you are managing some metrics are consistently collected and applied to your management approach.
Capability Level 3: Defined
Defined process is a managed process that is tailored from organization standard. Tailoring is done by justification and documentation guidelines. So for instance your organization has a standard that we should take invoice from every supplier. But for any instance if the supplier is not able to supply the invoice then he should sign an agreement and give in place of the invoice. So here the invoice standard is not followed but the deviation is under control.
Capability Level 4: Quantitatively Managed
Quantitatively Managed process is a defined process which is controlled through statistical and quantitative information. So right from defect tracking, to project schedules are all statistically tracked and measured for that process.
Capability Level 5: Optimizing
Optimizing process is a quantitatively managed process where we increase process performance through incremental and innovative improvements.

Figure: - Capability levels in continuous model
If you really see continuous representation is same as the staged only that information is arranged in a different fashion. The biggest difference is one concentrates on a specific process while the other brings a groups of process to a certain mature levels.