A cache is essentially a copy of active data that sits between the application and back-end storage, such as cheap disk drives. In contrast, storage tiering places data on multiple types of media (faster, more expensive versus cheaper, slower) depending on the activity of the data itself.

The point of read cache is to (a) prefetch sequential data and have next block ready for you before you ask for it and (b) store it for some time in the fast memory in case you’ll decide to read it again shortly afterwards. The point of write cache is to (a) rearrange random writes into series of sequential writes, (b) allow you to edit data before committing it to storage (if you write the same block twice, only one write operation will be performed on media) and © smooth down the write bursts/spikes into something more manageable by media. The point of tiered storage is to virtualize the storage. The data is stored on fast or slow media, depending on access frequency. It’s essentially the same as having system/software/games on SSD and videos/music/backup images on HDD, only you see just one disk of combined capacity and usually you don’t have to manage it, data moves between physical drives automagically.
And no, neither cache nor tiered storage will allow you to read data stored on “spinning rust” faster than it’s physically possible to read data stored on “spinning rust”.