This is a great answer. It brings into relief many of the thought processes and experiences which change in tackling questions after studying advanced mathematics deeply.

However, the answer seems mostly to cover what it feels like to study tractable problems. Things like reading and understanding others' research or work, or studying new questions which seem "within reach, or nearly so." This is representative because this kind of work forms the majority for most mathematicians, and nearly all of the work for non-professional mathematicians who have studied math deeply and keep abreast of their fields, but don't research full-time.

One aspect of the experience of understanding advanced mathematics which doesn't seem to be covered thoroughly here is what it feels like to study truly intractable questions, let alone those questions which you fear may actually be inscrutable. The simultaneous awe, respect and consternation one feels when confronting truly difficult questions which you intuitively feel you just don't have the tools for. The problems which you think you'll need to discover new tools to even begin breaking down. This answer generally projects confidence and fluidity in tackling problems. And that's fair, in that this is the biggest change one undergoes when tackling mathematical questions after studying advanced mathematics deeply.

For deeper questions-- those which you feel you are very far from being able to answer-- the experience isn't quite opposite of what is described here, but it is different. When you feel that all of your fluid mappings and transformations, all of the most powerful tools at the ready in your toolbox, and all of your simpler analogues are not only not going to solve the problem, but are unlikely even to lead to a truly deeper understanding of it... when you're not even sure whether breaking the problem down in a particular way will be productive or counterproductive... when you've wrestled with a problem for days, weeks, and months and you're still not sure if the "foundations and frameworks" you've built are even of the right type or in the right vicinity to solve the problem... then the sense of confidence projected in this answer falls away. The sense of assured solution-- even assured understanding-- and fluidity of movement in problem solving is no more. You're still confident in your mathematical knowledge and ability, and certainly you feel differently than you did as a beginner. But it is a humbling experience.

The process of tackling these questions is not quite so structured as the impression that might be given by reading this answer. In these situations, Wiles' analogy of stumbling in the dark through a great mansion for months and years (which the author also quotes) is closer than the analogy given by the author of building a house. What even Wiles' analogy doesn't quite capture, at least in the section quoted here, is the uncertainty of the process of stumbling through that mansion. At various points you ask, "does this room even have a light switch? At least one that I can reach?" There is tremendous backtracking as well, rather than the sense of steady progression in discovering and understanding room after room. Imagine stumbling through the dark for weeks or months, finally discovering the lightswitch in a room... moving to the next room and doing the same... then on again to the next room to do the same again for yet more weeks and months... then only to discover, based on what you've seen in these rooms, that you must in fact be in the wrong hallway. Or the wrong wing. Or even the wrong mansion entirely.

In any case, I think this is a wonderfully insightful answer overall and its author deserves great credit for being so thorough, so accurate regarding the great majority of work, and so descriptive and relatable. I just wanted to enhance the picture painted here, or expand one little corner of it, as it relates to other types of experiences one is almost sure to have in understanding advanced mathematics and trying to apply that understanding to work in the field.