It seems that when a mathematical theory was newly invented, or a particular phenomenon was discovered, it is often while tackling a specific hard problem, but as more of the theory was developed it found use in a variety of problems, many of which are "easier", as in not assuming as much prerequisite. Thus, when the textbook is written it is often the case that the author chooses to motivate with, or to focus entirely on, the easy cases.

To narrow down the question, I'd like to ask for examples in which the distinction can be phrased as finite- vs infinite-dimensional, at some level. To give an old example, the calculus of variations is almost as old as calculus itself, and it's basically optimization over an infinite-dimensional space. Its finite-dimensional analogue is certainly very useful, and is "easy" with differential calculus. It is said that the Lagrange multiplier was first discovered in the infinite-dimensional case—see epi163sqrt's answer to History of Lagrange Multipliers—yet it's now taught to all multi-variable calculus students.

Lie theory, Morse theory, Lax pair all seem to follow a similar history, though I'd like to read what the experts have to say. I hope it's of interest here beyond the purely historical, as it may be a better motivation for the theory when we are not constrained by the textbook-level detail.