Detergent will destroy the membrane much more than vinegar, and that's why the detergent leaks out more pigment from the beetroot slices in the detergent solution.
Detergent is an amphiphatic substance, meaning that it has hydrophobic (non-polar, electrons are shared equally) and hydrophilic (polar, electrons are shared unequally) ends. This means that it is an ideal surfactant and it is capable of reducing surface tension. When the detergent solution comes into contact with the plasma membrane, it forms micelles (little globules of fat formed from the hydrophobic fatty acids of the phospholipid bi-layer) and thus tears apart the plasma membrane. As substances can only diffuse through the membrane when gaps are created in the membrane (because of the fluidity of the membrane), a lot of pigment will leak out of the membrane due to the bigger gaps in the membrane, and a lot more substances can diffuse into the cell.
The vinegar solution is acidic and contains more hydrogen ions than hydroxide ions. Thus, amino acids will tend to donate or accept protons in order to stabilise pH levels in the cell. Since the amino acids have changed their structure, this leads to a change in intermolecular bonding between amino acids, in particularly, hydrophobic interactions and hydrogen bonds. As a result, the secondary and tertiary structure will change and the protein will denature (because it's changed shape, it cannot no longer carry out its specific function). Pigment will leak out of the cell as a result BUT the phospholipids in the plasma membrane are not affected, thus no bigger gaps are created in the membrane and less pigment leaks out.
*Your SAC answers do not need to be this long
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