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Same here. If one did not believe that an outcome (aim) was actually an opportunity... does not compute.
You apply to a bunch of jobs, get a few interviews, a couple of offers, pick one. That’s response to opportunities. You decide you’re going to get to Mars or prove the Poincaré conjecture with no clear sense of whether there’s an opportunity to do so. That’s a goal.
Are you trying to point to the distinction between present(ed) opportunities and those which are contingent upon one's capabilities at a future (uncertain) date?
Roughly. It’s a loose distinction, not a fundamental one. An opportunity is access to a leveraged pathway to a valuable end. It’s easier but also more fixed.
Decided to start a company during college based on argument from my friend "we will never have better access to intellectual capital or more risk tolerance than now". Does that count?
Well looks like you did, so yeah 
Learning math for real after school let me down
Why not both? I take opportunities for predictable subsistence. I also have goals that require comfort with uncertainty and decades of abduction/deduction cycles to find the practical path. It takes passion and persistence.
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