If you hire a consultant, you ask them what you should do. If you hire a contractor, you tell them what they should do.

There's quite some overlap, but generally a consultant is more specialised and has a higher billable rate and works on shorter term projects than a contractor.

e.g. my company currently has a couple of Navision consultants doing integration work for our invoicing system. Once the project is done they'll probably leave. I imagine they bill a lot per hour. Contrast to me (contractor) working on multiple projects, and I've now been there for 1.5 years.

Contract workers are "contractors". There is a clear deliverable and the worker produces it.

Consulting is the practice of engaging a business/org/lob to address a larger, less clear-cut problem. M&A is for M&A Consultants who specializes in acquiring and merging. Out of the merger may arise a unit of work to be "contracted out" in cases where timeline and capability are out of sync.

Can you consult under contract? Sure, but being a contractor doesn't automatically make you a Consultant.

There is a gray area but typically you would hire a contractor to address a bandwidth issue and fill an already well defined role in your org. You would hire a consultant to address a capability issue: they will typically know more about the area than you do.