Thanks for the question, Moira! “Contractor” normally means someone who works under contract, which is how most of us work: we sign a contract to perform specific work/deliver specific work products (like a building, a report, a new device, accounting services, etc.). This is how we get paid. “Consultant” simply means an expert or semi-expert in a topic area who is consulted for advice. They provide services, rather than hard products (e.g., travel demand forecasts for a region, rather than a new bridge over a river). They also work under contract. Thus, a contractor is more broad than a consultant.
Many people think of construction work when they hear the term “contractor”, and you may have that idea in mind too! Those who are contracted to construct something use math (for shortest-time construction paths and net present values of project investments, for example), but not as much as those contracted as consultants on other, more complex civil engineering topics (like traffic forecasting 20 years forward, wave speeds in an ocean against a port facility, the chemistry of air pollution formation, earthquake forces on rooftops).
Civil engineering is an incredibly diverse field, from water and air quality to structural strength, materials hardening to transport logistics, bridge construction to soil liquefaction, and much more. You can do as much contracting and math as you want (or don’t want). We look forward to having you in the discipline! Good luck with your coming studies.