the conciseness of NFAs
If NFAs are no more powerful than DFAs, why use them? For the same reason you sketch a rough plan before drawing the blueprint: NFAs are often far smaller and far easier to design. A property that takes a clean, small NFA can require a much larger DFA — sometimes exponentially larger. Nondeterminism lets you describe WHAT to look for and postpone the messy bookkeeping of remembering exactly where you are.
The classic illustration is the language "the k-th symbol from the end is a" over Σ = {a, b}. An NFA solves it with about k+1 states: stay in the start state guessing you are not yet near the end, then GUESS the position of that k-th-from-last symbol, check it is a, and count off k-1 more symbols to the end. But a DFA must, in effect, remember the last k symbols read at all times, so it needs on the order of 2^k states — one per possible window of recent symbols. The tiny nondeterministic guess replaces a large deterministic memory.
This succinctness is the real, practical payoff of nondeterminism for finite automata. It is honest to repeat the caveat: the conciseness is purely about description size, not about computing more. Any NFA can be turned into a DFA by the subset construction, and although that DFA may blow up exponentially in the worst case, the LANGUAGE recognized is identical. NFAs save the human effort of design and often the page count; they do not extend the boundary of what is computable.
"Third symbol from the end is a" (k = 3): an NFA needs only 4 states (guess where the third-from-last symbol is, then read three). The smallest equivalent DFA needs 2^3 = 8 states, one for each possible pattern of the last three symbols it must remember.
A k+1-state NFA versus a 2^k-state DFA for the same language — conciseness, not power.
Conciseness is about description length only. The NFA and its DFA recognize the SAME language; nondeterminism never lets a finite automaton recognize a non-regular language.