Trademark, the most formal usage, is used for trademarks with legal effects, such as registered trademarks. Brand, trademark, more refers to the brand, such as Coca-Cola. The verb usage is to brand a trademark. Logo refers to the logo of the trademark, which is a symbol or text. For example, Nike's swoosh is the LOGO. It focuses on that logo, that graphic. And mark means making a mark. Compared with label, the mark made by mark is more casual. If you mark your notes in class, you'll be in trouble. They're all messy, haha.

Label also means making a mark, which means "adding a label". But it's a bit more formal than mark. For example, if you want to make a mark on a country's official documents to distinguish them, you would use label instead of mark. Brand name is a brand, trademark, product name; brand is a brand size relationship, brand>brand name.