Material Review Board is coming under Quality Assurance (a precautionary action). Control Plan is coming under Quality Control (a controlling action).

In simple terms, if you want to check a predefined specific property or a measurement of your parts in different stages such as Manufacturing, Purchasing and Inventory go for Control plans. It is a reliable feature which you can turn on and can’t avoid checking in some stages of your part lifecycle. Further you can implement ISO standards for quality, using IFS Control Plans.

MRB from quality assurance side, you can turn on whenever you want and use to process non-conformance material. Non-conformance material is defined as any raw material, part, component, or product with one or more characteristics that depart from the specification. Such material can be discovered or inadvertently created during the manufacturing process, inventory handling, or purchase receipt process.

Further, using MRB you can approve, return to the supplier, rework or scrap the reviewed material partially or completely. For example, if the total number of parts to be reviewed, is fifty, all fifty parts can be approved, scrapped, returned or reworked. Otherwise only twenty-five parts can be approved and the remaining twenty-five can be scrapped.

As a real example, I can add a control plan and measure the diameters of my manufactured pots in shop order manufacturing level quality controlling factor. If the diameter values, go beyond the limits I can go for a rework or scrap. If not, I can approve quality and ship the pots.

While I’m doing my manufacturing process, I notice some color change in my pots. Hence, I create a MRB at the Shop Order level and check for material property or color. If the test has been passed, I can ship the lot or rework, scrap.

Same thing in Purchasing and Inventory levels also.

As a summary, control plan is predefined and which we don’t avoid. MRB is something that we turn-on a request (on demand).

Control Plans are set up to create Analysis where you register your results. It could be you measure something or a visual inspection. The Analysis results can later use in SPC (Statistical Process Control) charts as well where you can see the trends and take necessary actions before things get worse.

MRB is different, where you create an MRB case when you come across a nonconformance, where you can’t take an immediate decision and need some expert advice on what to do.

Eg.  Say as a control parameter you check the Color of your product. If the Color is quite OFF than the expected color you can straight away scarp (depend on the business). But in a situation where you can’t make a decision, you register the Analysis result as not within specification and create an MRB and assign it to relevant staff to carry out the MRB. After a further inspection of it, MRB will decide what to do. It would be use as it is, scrap or rework or send back to the supplier, etc. This could be crucial especially dealing with Purchase material that you need some experts’ opinion on what to do.

As a summary, Control Plan to create an Analysis where you can register your observation or measurements, while MRB makes decisions case by case on what to do.

Just as a small addition to the already existing answers.
In my customer projects, I always use the NCR (Non Conformity Report) instead of the MRB. The advantage is that it is much more flexible and can be attached to any desired object (and not only to receipt, shop order or part in stock). I would always recommend the NCR to manage non-conformities.