There's no foolproof way to measure experienced pain, since everyone experiences it differently and has different tolerances. However, experiments like the one you describe can give us some insight into how different people react to pain. Additionally, medical professionals often use pain scales (like the 0-10 pain scale) to get an idea of how much pain a patient is in. This in the future can be used to help develop new pain medication or ways to cope with pain. However, when new technology like for example a neural scanner is used to measure pain, we will learn much more about how different people experience pain and what methods work best to treat it.

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As to if there are differences in how physical and mental pain is experienced, there is some evidence to suggest that they are indeed different. For example, people who have suffered from physical pain often report that it is more "intense" than mental pain, while those who have experienced mental pain often say that it is more "distressing" than physical pain. Additionally, mental pain is often more long-lasting than physical pain.