Why can't you just copy a qubit and check the copy later?
Classical data has no problem with this: copy a file a billion times, verify each copy independently, keep the majority answer. It's the basis of essentially all classical error correction.
A qubit in an unknown state cannot be copied that way, for reasons that have nothing to do with current technology. If it could, quantum key distribution would be trivially breakable (intercept, clone, measure one copy, forward the other undisturbed), and quantum error correction would look completely different, built on backup copies instead of the far stranger trick it actually uses (spreading information across entangled qubits so that no single qubit alone ever holds the answer).
The catch, and it matters: this only forbids cloning an unknown state with a single, universal procedure. If you already know what state you're preparing, you can obviously prepare it twice. What you can't do is hand a machine one copy of a state nobody told it, and get two.
Claim. There is no unitary U and fixed blank state |0⟩ such that U(|ψ⟩⊗|0⟩) = |ψ⟩⊗|ψ⟩ for every state |ψ⟩.
Proof. Suppose such a U exists, and suppose it works for two states |ψ⟩ and |φ⟩:
U(|ψ⟩⊗|0⟩) = |ψ⟩⊗|ψ⟩
U(|φ⟩⊗|0⟩) = |φ⟩⊗|φ⟩
Unitary operators preserve inner products, so taking the inner product of the left-hand sides must equal the inner product of the right-hand sides:
⟨ψ|φ⟩ · ⟨0|0⟩ = ⟨ψ|φ⟩ · ⟨ψ|φ⟩
⟨ψ|φ⟩ = ⟨ψ|φ⟩²
Let x = ⟨ψ|φ⟩. The equation x = x² has exactly two solutions: x = 0 or x = 1. That is, |ψ⟩ and |φ⟩ must be either orthogonal or identical. So a single U can clone at most one direction's worth of states, never two genuinely different, non-orthogonal ones. No universal cloner exists. (Wootters & Zurek, Nature 299 (1982); independently, Dieks, Physics Letters A 92 (1982).)
Two things this site already teaches, both explained by this proof Quantum error correction cannot work the way classical error correction does (no backup copy is possible) which is exactly why it instead spreads one logical qubit's information across many physical ones via entanglement (topic 02 above is the tool that makes precise what that spreading even means). And quantum key distribution's security (topic 03) has a second leg to stand on beyond distinguishability: even if an eavesdropper somehow captured a qubit in transit, they cannot clone it to measure at leisure while forwarding an undisturbed original. They get exactly one shot, on the one photon that exists.
Check yourself
The no-cloning proof ends at the equation x = x squared, where x is the overlap of two states. What does that force?
x = x squared has only the solutions 0 and 1, so a single unitary can clone at most states that are orthogonal or identical. No universal cloner exists.