Could you provide a reference that homotopy equivalence which pulls back tangent bundles is a diffeomorphism? Does it somehow follow from the H-cobordism theorem?

The point I'm trying to make is that the term "classification" is often used once a (reasonably computable) complete set of invariants has been found. Freedman's classification of 1-connected (smooth) compact 4-manifolds up to homemomorphism depends solely on the intersection form, a readily computable object. This is considered a "classification" regardless of how easy it is to compare 2 intersection forms to see if they really are the same. In short, often "classification" means "reduced to a problem in another field of mathematics", even if this new problem is (known to be) hard.

I wanted to add something slightly more exotic. $SO(8)$ is diffeomorphic to $SO(7)\times S^7$, but $S^7$ doesn't have ANY group structure (though it ALMOST does: there is a multiplication which satisfies all of the usual group axioms EXCEPT associativity. Google search Cayley numbers for more information).

I agree (especially that "classification" is somewhat vague), but there is a difference between a computable invariant and comparable invariants. For example, the fundamental group is "reasonably computable" via van Kampen's theorem, but telling when 2 groups are isomorphic is algorithmically hard. Thus, given a manifold M, I can output the fundamental group, but given 2 manifolds, I cannot (algorithmically) distinguish them via their fundamental groups. That said, I think almost anyone would be willing to accept "fundamental group" on any list of "reasonable" invariants.

Well, the real line acting on itself by translation makes up the identity componenet of the isometry group in this case - but the full isometry group is disconnected. For more examples like this, if you pick any connected centerless Lie group G at all and pick a "generic" left-invariant metric, I'd be willing to bet that the identity componenet of the isometry group is isomorphic to G. (But I don't know this for a fact)

Someone edited this post, and in an effort to see exactly what was changed, I accidentally reverted it to it's current form. For whomever edited it, if you'd like to re-edit, feel free.