You can't rely on them existing, having space, or keeping your data for any real amount of time. You can't process the data before it hits the cache, you can't clear them easily, and they are iffy at best when offline.
In other words, a cache.
If I do an expensive operation, then need to use its result later, I can't rely on the HTTP cache as the browser may never have cached it in the first place, or may have evicted it by the time I go to use it. So using that unreliable cache can actually make things worse in the sense of more data usage, more power usage, more battery usage, and more memory usage.
For a real world example, say I wanted to pre-fetch a bunch of data on page-load, then in a later page I want to use the cache to be able to quickly get that information and use it without any latency. If I had some control over the HTTP cache, I could do this.
But as it stands right now, I can try to make a GET request on the page load, but when the user clicks the button to see the result on a later page, it's a crapshoot on whether or not the HTTP cache will still have it, leading to the user having to wait sometimes, and not other times, duplicated requests, and a lot more data usage.
With localstorage, IndexedDB, WebSQL, etc... I have that control. I can know that the information is going to stay there until I need it so I don't need to worry about my caching making things worse.