I am looking around here for pics of gardens. I discovered the Cottage Garden Forum in June, and post there a lot. When I first saw this, I recognized the flower someone had planted in a bed I took over gardening at church. It was my favorite flower in the patch, and then someone informed me I was growing a plant that is illegal to grow. I was very sad. I now have a couple of Russian Sages that love the area, and get very full, too. Sue

Looks like a type of Alocasia. Here is a link that might be useful: pictures

Do they smell or taste different? There's no real way to tell from the photos. In my experience, the way your farmer designated them is typical for the way each of them look. But it's quite often the other way around, and it's also quite often that they're both the same. Peppermint leaves are supposedly quite a lot darker than spearmint leaves, but I can't really see all that much of a difference from your photos. Some people will tell you to look at subtle differences in the teeth of the leaves; others say that peppermint stalks are always purple, never green; some people say that the flowers are arranged differently. Most of that is relative to form, in my opinion. There are countless variations of mint commonly sold under names like "mint", "spearmint", "common mint", "garden mint" and sometimes plants that look like "peppermint", and are sold under that name, are actually just another mongrel form of spearmint. You can buy packets of "peppermint" seeds from almost any nursery, and what grows will look a lot like a typical form of peppermint, but since peppermint is sterile, there's no way that it's actually peppermint. I guess the only definitive way to tell is by testing the oil composition, which you can do to a degree by taste and smell. I don't really know to describe the difference without resorting to confectionery, but it should be pretty noticeable. I was going to also say that if it's spearmint, you should be able to propagate it from seed. But a lot of people never see flowers from their spearmint so that's not likely to work as a test.

Just google the scientific name and see what turns up. A quick look found a plant for sale here: https://www.burncoose.co.uk/site/plants.cfm?pl_id=1740.