Probably the question that should be asked is if the result can be obtained from scratch without knowing it! Often, many textbook or homework problems ask students to use induction to prove some theorem, but the student would be unable to come up with the theorem in the first place. This kind of problems is no good. For example, instead of asking for a closed form for $\sum_{k=1}^n k^2 2^k$ (using only arithmetic and exponentiation), they give the answer and ask for a proof by induction, which is almost totally useless.
One can see a sharp distinction between proofs by induction where the hypothesis "drops from the sky" and proofs that naturally arrive at the conclusion by other means. Despite the fact that all the other proofs must use induction at the formal level, they feel more natural when there is no strange and inexplicable cancellation of terms that occurs when you prove the answer correct by induction.
As t.b. commented, in combinatorics one would generally prefer bijective proofs basically for this reason; nothing cancels and there is a clear direct correspondence between one collection and another. Similarly it is better to show how to solve a recurrence relation by a general method than to just state the solution and prove it by induction.
Such considerations of cancellations in proofs occur in other ways besides the use of induction. In geometry one prefers a synthetic solution (using geometric arguments) rather than an analytic solution (using equations over a real-closed field), simply because in an analytic solutions there is usually unexplainable cancellation of factors along the way that never show up in a purely synthetic proof. Some even avoid trigonometry for the same reason, but some theorems are far nicer when expressed in trigonometry.
The amount of redundancy in a proof, corresponding to unnecessary detours that result in later cancellation can be somewhat quantified in Euclidean geometry by the total degree of the rational functions involved. In the most elegant synthetic solution one almost always deals with linear or quadratic expressions and there are no unnecessary cancellations, but in an analytic solution every circle intersection doubles the degree by two. In fact this is why automated theorem provers still cannot handle complicated geometric theorems that have reasonably short synthetic proofs, because humans can construct new points that link existing ones in a simple way that avoids later cancellation.
In logic we have the cut-elimination theorem for sequent calculus, which is a bit like saying there is always a direct proof that does not use any new idea that is not already contained in the theorem to be proven. However, as proven by George Boolos, not allowing such indirect proofs (using cut) can force the minimal length of proof to be far larger than otherwise. So if we go by proof length to compare proofs we may need some 'cancellation'.
There is therefore a trade-off between proof length and explanatory power. In the example I started with, the shortest proof will probably be by induction! But the most satisfying one will show how to solve such problems in general, through the anti-difference operator and anti-difference by parts. It is quite clear that here structural insight is more important than a short proof, since the 'new ideas' introduced do not just solve this problem but a whole class of problems. Furthermore, these enable finding the closed form without knowing it. This reminds one of the distinction between P and NP, where a problem in NP has a solution that can be verified in polynomial time, exactly like most textbook induction problems can be verified in a routine way, but a problem in P has a solution that can be found in polynomial time.
Anyway these are just my views, since I also have thought much about this kind of issue with induction. Some instances of induction are so natural that people do not even notice it and sometimes insist there is no induction, like whenever you use a summation sign. Other times induction feels like the wrong tool. One such simple example is the handshake-lemma in graph theory, where there is an induction proof that feels like it does not explain anything, and the double-counting proof that explains it all. Many students claim that the latter does not use induction, but formally it does!
