First, let me recall some details for readers unfamiliar with the topic:
The (classical) ω-rule intuitively says: if you can prove the sentence "P(n)" for each specific natural number n (conflating the number n with its corresponding numeral for simplicity), then you can prove the sentence "∀xP(x)." It takes a bit of care to make this precise and non-tautologous. Ultimately, what we have is a definition of a class of things called valid sequents, and the ω-rule is one rule among many (the others being the usual rules of the sequent calculus for first-order logic) which says the following:
If Γ ⊢ P(n) is a valid sequent for each specific natural number n, then Γ ⊢ ∀xP(x) is also a valid sequent.
(Think of "Γ ⊢ p" as meaning "From Γ we may infer p." The distinction between ⊢ and → can be hard to see at first, but it winds up being important; for now, just take it for granted.)
The set of ω-valid sequents is then the smallest set of expressions of the form "Γ ⊢ p" (for Γ a set of sentences and p a sentence in the language of arithmetic) which is closed under the sequent rules, including the ω-rule.
By contrast, induction - which I'll phrase as an inference rule here, to more closely compare with the ω-rule - says the following:
If Γ ⊢ P(0) is a valid sequent, and Γ ⊢ ∀x(P(x)→P(x+1)) is a valid sequent, then Γ ⊢ ∀xP(x) is a valid sequent.
Note that the ω-rule treats all instances equally and involves no high-complexity expressions, whereas induction treats 0 as special and involves a high-complexity expression (the "induction clause"). So there is indeed an apparent discrepancy between induction and the ω-rule.
OK, now on to the question. As is so often the case, I think it's helpful to respond with a new question:
What motivates the ω-rule in the first place?
Certainly we don't believe it in general - there are lots of discrete ordered semirings where it breaks down. Something about ℕ specifically means that it's enough to check every "concrete term" (think Herbrand semantics) in order to verify that a property holds globally. So where does that come from?
The answer will depend on presentation, of course, but broadly speaking we're looking at the well-ordering principle: ℕ is assumed from the get-go to be "minimal" in an appropriate sense. Specifically, no proper subset of ℕ can contain 0 and be closed under successor. But this exactly matches the form of induction! So really the ω-rule is a consequence of a "higher-level" (= set-theoretic) principle which does parallel induction.
This, I think, does partially address your question by explaining away the apparent difference in form between the ω-rule and induction. However, there's still the question of effectiveness of the latter. At this point it's worth observing that induction is actually parameterizable: rather than restrict the induction scheme to formulas of first-order logic, we can consider "𝔏-induction" for any logic 𝔏, e.g. second-order logic, fixed-point logic, ... See here for more on this idea.
So in a sense, the idea of an "internal induction scheme" is really setting the stage for a gradation of induction schemes. We can now consider the following spinoff of the "strength" part of your question: why is first-order induction so "close to the top" in the parameterized-induction-scheme hierarchy? This actually leads to some interesting apparently-open questions, but one bit of evidence is that any "reasonable" logic 𝔏 whose induction scheme is strictly stronger than that of first-order logic in fact pins down ℕ up to isomorphism. See e.g. here. The narrative that leaps to mind, then, is that "natural" properties of ℕ should be implied by fragments of the induction idea given by "reasonable" logics which don't require perfect information about ℕ, and that limits us exactly to first-order Peano arithmetic.