There are three important keys to any project:
Part of project management is figuring out how to balance those priorities properly for the specific project.
Cost plus is a favored billing strategy for when the most important thing is getting a completed project especially where you need flexibility to make changes mid stream, and the work is going to be done by an external team. This may be an team in another part of the company, or less commonly an external vendor. It is also a favored method for contract companies when dealing with the government due to shifting requirements, and payment on deliverable requirements.
The upside here is the flexibility to make changes to the project midstream with out having to renegotiate the contract, and often it is only payable if the deliverable meets the requirements, so there is no out of pocket cost if the project fails to deliver.
A big down side is there is no incentive for the consulting company to keep costs down. In fact they are rewarded for doing the most they can to meet requirements.
This is a technique where the final price and timeline for the project is set before work on the project is started.
If you have a good negotiator and a solid set of specifications for the project this is the route to go. The down side of this method is there is little to no room to make changes in the middle of the project.
This is basically hiring a temporary employee with out the overhead and hr issues. Typically the consultant is managed by someone in the contracted company but they work to your direction. The work they do belongs to your company.
This is great if you have strong project management but are lacking some specific skillset and/or manpower for a project. This can be a short or long term engagement where the consultant does work as part of your team, and you pay a set hourly rate based on a contract.
The down side here is you are responsible for managing the work of the consultant and making sure they are productive. You can reduce their productivity greatly by loading them up with meetings, and overhead tasks(like reporting, answering email, and required training). But when well managed a consultant can be a huge benefit to a time crunched project.
This is similar to an internal consultant with the exception that the work is done outside of your office. Typically an external consultant will not bill for the overhead tasks like email, staff meetings, etc. But the consultant is also managed by the external company. It also allows the company to allocate its resources to best fit the needs of your project. So you may be able to get the services of several different professionals on an as needed basis.
This has the draw back of you are allowing the consulting company to manage how the work is being completed. These resources are not available to your company to pull into meetings, or help on an emergency. The rate for this type service is also usually higher because you are only being billed for the time they work on your stuff. But if your office has a problem with impromptu meetings, and other things popping up that push timelines out, this can help keep a project on track as well.