both an engineer and an entrepreneur are compelled to solve a problem, it seems though that difference is the entrepreneur is also compelled to profit financially from the solution.
Engineers profit financially too. Anyone that's acqhired is leaving entrepreneurship for engineering for the promise of a big financial payoff.
I'd say the main difference is that an entrepreneur "shifts resources from areas of low productivity to high productivity", while an engineer comes up with the most viable technical solution to a problem. In other words, an entrepreneur's job description includes opportunity costs, while an engineer's job description leaves that for an executive or product manager to worry about. Profit is a natural consequence of that, since if you are successfully shifting resources from areas of low productivity to high productivity, the difference can be captured as profit.
I'd say the main difference is that an entrepreneur "shifts resources from areas of low productivity to high productivity", while an engineer comes up with the most viable technical solution to a problem. In other words, an entrepreneur's job description includes opportunity costs, while an engineer's job description leaves that for an executive or product manager to worry about. Profit is a natural consequence of that, since if you are successfully shifting resources from areas of low productivity to high productivity, the difference can be captured as profit.