They are both specific to the drive, not generic parts so their exact contents is only known to the manufacturer but my best guess is the smaller IC is the motor controller, it would generate the phase signals for the spindle motor and probably the voice coil drive signal too, in other words the mechanical parts of the drive.
The bigger IC will be the data processor, analyzing the data to be written and calculating the bit patterns to be send to the heads, including CRC and early/late compensation. For reading it would check the sector headers so the head position could be confirmed, check the sector CRCs and frame the data from the heads into a parallel format for the PATA bus interface.
Generally, drives do not understand things like directories/folders and files, that would be done by software or another device on a motherboard, that is what is usually referred to as the HDD controller. Drives are just a storage device, they have no notion of what is actually being stored and where.
Brian.