An advisor gives some kind of advice. He can do it as a job, or he can do it informally. You can have a professional political advisor, who gives advice to politicians, a financial advisor (who can work for a company or have his own practice), or many other types of professional advisors. But an advisor does not have to be a professional. Very often, a typical immigrant to the United States has a friend or relative who tells him what he should do, how to stay out of trouble, how to handle his money, etc. That person is also an advisor, because he gives advice.
A consultant is someone you consult. You can consult him for advice (in which case he is also an advisor), for technical services, or for other kinds of work.
In the good old days, a consultant was usually someone with years and years of professional experience. He then went out on his own and rented his expertise to various clients. Now, however, there has been a sort of title inflation, and some companies call any freelancer or outside contractor a “consultant”. So, you get the strange situation where a kid right out of school, with no experience, is called a “consultant” just because he is freelance and the company doesn’t withhold taxes from his pay and doesn’t pay for his insurance. It’s a little bit like companies where everyone is some kind of vice president.