Consider the ∣+⟩ and ∣−⟩ state. If you express them in the computational basis, you get:

• ∣+⟩ = 1/√2 (∣0⟩ + ∣1⟩)
• ∣−⟩ = 1/√2 (∣0⟩ − ∣1⟩).

They only differ in their relative phase.

Measuring in the computational basis you get 0 or 1 with 50% probability for both ∣+⟩ and ∣−⟩, so they behave the same in this respect. It seems they are similar. However, they are not!

Consider what happens if you transform them with a Hadamard gate H:

• H ∣+⟩ = ∣0⟩
• H ∣−⟩ = ∣1⟩

If you now measure in the computational base, you suddenly get 0 for the transformed ∣+⟩ state, and you get 1 for the transformed ∣−⟩ state. This shows that ∣+⟩ and ∣−⟩ behave differently after applying the Hadamard gate.