In my view, freelancing is a subset of consulting. Consultants design, and in many but not all cases, implement solutions to business problems. Freelancers implement solutions already designed by others.
Consulting generally pays more because the value lies in creating the solution to the problem, not in commoditized coding time. Clients typically know the problem they have before they call in a consultant. It is less common for them to have the solution completely mapped out.
A contractor is a replaceable cog that does what they're told (though might be an expensive cog). What you're incorrectly calling a freelancer. They get hired at a daily rate and turn up at work and told what to build.
A freelancer will usually have complete autonomy on the tech side, but no say in the business side. A freelancer is not always expensive. Freelancers are sometimes cheap consultants. A guy who builds WordPress sites for mom & pops is a freelancer. They were asked for a website, not a WordPress one. Bit also a guy who makes mobile apps is a freelancer, but it's his choice to choose swift or not.
A consultant is there to bring specific knowledge about a specific business problem or metric. They are always expensive.
In many cases, freelance jobs meet your definition of a contractor. I suppose that is the problem with this whole argument, as these definitions are somewhat subjective, and the lines between them are crossed all the time depending on how desperate for work the consultant/freelancer/contractor is. That's why I prefaced my comment with "in my view". The only real consensus is that consultants are at the top of this particular food chain.
Consulting generally pays more because the value lies in creating the solution to the problem, not in commoditized coding time. Clients typically know the problem they have before they call in a consultant. It is less common for them to have the solution completely mapped out.