How is data physically written, read and stored inside hard drives? Is it stored as differing spin of atoms inside the drive? Is that method the same in flash drives and SSDs?
Hard disks use magnetic storage,
i.e. the information is stored in the South/North orientation of small pieces of the magnetic medium (also the case for tapes in the old tape recorders; and credit cards; these applications differ by the "geometry" how the bits are arranged but not by the essence how a bit is stored). A stronger magnet in the head is able to remagnetize the pieces of the surface; a weaker magnet is able to "feel" in what way the pieces are magnetized.
USD flash disks and SSDs use flash memory,
which evolved from EEPROM memory chips. Flash memory is composed of flash memory cells which primary contain the floating-gate transistors; such cells have a control gate (CG) and de facto isolated floating gate (FG). FG can keep electric charge for years and the electric field from that affects the threshold voltage of the cell (voltage at which the cell becomes conductive). Some extra operations are needed to erase and rewrite the chips which can't be done indefinitely; after less than a million of erasures, the cell breaks down.
There are many other physical mechanisms in which the information is being stored. All the "relatively easily" rewritable technologies use some kind of variable electric or magnetic fields that may be produced by matter. Compact disks (CD/DVD) use optical information and so on.