What Yuval wrote is correct, but that is more about formal proofs and from the perspective of a logician or a person working in foundations of mathematics. I want to explain one of the reasons that people sometimes claim that a non-inductive proof is better than another one which is explicitly using induction.
From formal and foundational perspective, you may need to use induction to prove the statement working a formal theory, it might be there explicitly or it might be hidden behind lemmas and theorems that are being used.
So why is it sometimes claimed that a proofs is better than another one?
Because a proof is not always a formal proof (an informal proof is something that would convince you about the truth), and because a proof contains more information than just the truth of the statement. It tell us why the statement is true. This is not a rigorous (AFAIK) but rather an intuitive one. Mathematics is not just formal proofs, intuition is also an important part of it. Over time we learn the skill to understand some mathematical concepts, objects, theorems, ... so well that we don't need to check their formal definitions or proofs anymore, we start to "see" them (some can see a reference to Godel's views about philosophy of mathematics here :). And when we "see" them, we don't need a formal proof for them to use them.
Sometimes when we work with Yuval on a topic that he is more knowledgeable than me, he claims some statement is true and I have no doubt that the statement is true but I don't see that it is true at first. I don't dispute the truth of the statement but I tell him "I don't see it", and he explains it more and then I also start to "see" it! :)
From the perspective of a beginner that does not see the truth of any mathematical theorems and needs proofs for all of them (which from foundational point of view will need induction) it might be the case that there is not a big difference between the proofs. But you hear a lot when some mathematician claims that one proof is better than another one. The main reason is that a proof helps us intuitively understand the reason a statement is true, it helps us "see" that the statement is true. It is more than just expressing that the statement is true. Different proofs give us different perspectives on its truth. A completely formal proof as a sequence of formal mathematical expressions can show the correctness of a statement, and we can check that the proof is correct, it is a mechanical task of low complexity, but often it will not tell us the reason the proof works, it does not help us understand the reason the statement is true. On the other hand, a better informal proof using concepts and theorems that we "see" can help us in understanding the reason the statement is true, and hopefully eventually we might even "see" that the statement is true.
Using induction can be similar to doing a formal proof, while using other concepts and theorems about them is similar to the informal proofs that use what we can already see.