Honestly, there's a lot of confusion (and overloading, to use OOP parlance) with the term "contractor", and confusion with the term "consultant".
For instance, you can be a "contractor" as a software engineer, where you sign an employment contract to work for 6 or 12 months at a client site, working through an employment agency. At the end of that term, either your contract is renewed or it is not and you're now unemployed.
Or, in the defense industry, you can work as a "contractor" where you're a full-time employee of some company (called a "defense contractor"). That company signs a contract with the government to do some work for them somehow, and you're assigned to do this work. Your work does not have a defined end date (like the 6-month contract you might get to work at some big commercial company), you're just a regular full-time employee, but you still have to track your time (so it can be billed to that government contract), and if something goes wrong with the contract (like it runs out of money), then you can suddenly, with little warning, be furloughed without any pay.
Of course, there's many other kinds of "contractors" too, like the guy you hire to replace your house's roof. And there's all the people who are full-time workers, but are given 1099 forms as classified as "independent contractors" so their employer doesn't have to pay FICA and unemployment insurance for them.
Now, as for "consultants", that usually seems to be pretty much what I described in the part about defense contracting: you're a regular W-2 employee of some corporation, but you're sent out to client sites to do work for them, though there's no fixed-term contract with the client usually.