When you delete your browser history, you’re simply deleting a record of the sites you’ve visited. This will prevent them from showing up as you type in the address bar, and ensures they won’t be visible when the history menu is opened.
Cookies are completely different. These are small pieces of data that a websites stores on your device in order to keep you logged in and enhance your experience in some way (or, as is increasingly common, to follow you around the internet to better target you with ads). Clearing these will require you to log back into most websites, but will stop sites you haven’t visited in a long time from continuing to see what you do online.
And finally, there’s the cache. Essentially, when you access a website, your device saves parts of it (such as images) locally, so that the next time you visit, the page will load a little bit faster. Over time, the cache can swell, and it’s a good idea to clear it every once in a while. Doing so ensures you’re not storing data from sites you no longer use, and can help fix the bugs that can occur when a website updates and no longer matches the version stored in the cache.