For many, these terms are interchangeable. For us, however, destructible means that the label breaks into tiny pieces upon attempted removal. Tamper evident labels are either destructible labels or labels that show a void or checkerboard pattern when you try to remove the label.
We consider tamper-proof and tamper-evident to be equivalent terms. However, tamper-resistant labels or tags can help stop a vandal from altering the information on the label. Self-laminating labels and tags are good examples of this kind of system. Once laminated and sealed, your writing and printed information is difficult to alter.