The key to consultant vs contractor would for me the focus on specialized advice with the former, but that doesn't mean it is just that. E.g. a consulting gig could start with clear consulting ("talk to the customer about their problem, investigate, give them an expert opinion on how to solve it") and then turn into the consultant also doing the implementation. But a consultant also could just produce a report, or direct the client's employees on how to solve a problem.
And contractors will often do consulting in the beginning stages, e.g. if they're hired to "build a website", they might give the customer specialized advice on what to consider or prioritize, ...