Are mint and peppermint the same thing? Can we substitute one for the other. How can we differentiate between mint and peppermint?
Peppermint is a hybrid breed of two plants belonging to the mint genus, spearmint and watermint. In my experience, when 'mint' is referred to by itself without any other descriptors, it usually refers to the spearmint flavour people are used to (from things like green restaurant mint candies, toothpaste, etc). Peppermint will be denoted as peppermint. However, as mentioned in the comments, this may vary based on region.
There is a significant difference in taste between peppermint and spearmint; I find peppermint to have a much more intense flavour. Scientifically, peppermint gets its flavour from its high menthol content, whereas as spearmint owes its flavour to the compound L-carvone. Substitution will not replicate the same flavour as the original.
Source for science related parts: https://en.wikipedia.org/wiki/Mentha
Mint is the genus of the herb, peppermint and spearmint are two subsets. Spearmint is milder and more classically European, peppermint is rather more aggressive, and much more popular in the US (where it's usually labelled as just 'mint'). Those are the two main subsets.
You can substitute one for the other, but you will lose the characteristic flavor of the dish if you're aiming for something specific. For instance, mint tea made with peppermint doesn't taste Moroccan at all :-). If only 'mint' is specified, you can use whichever one you prefer (it's very easy to grow, so you can have your own right in your kitchen).
The only way I know to recognize them is by smell, so you have to find both different sorts and learn to distinguish them by experience.