The CY2305-1H is the high-drive version, and its rise and fall times are much faster. As shown in Switching Characteristics for CY2305SC-1H in the datasheet as compared to the CY2305SC-1.
The -1H version also has a tighter Duty Cycle specification for frequencies below 50 MHz. The -1H version also has a minimum slew rate specification (t8 measurement).
These are the key differences. In general, the -1H version was created to meet tighter timing specifications or to drive larger loads.