The 74HC has different logic levels than plain 74 and 74LS. It's also incompatible with 3V logic, when operated from 5V.
The 74HCT has the same logic levels as 74(LS). It's useful for shifting 3V logic up to 5V and interfacing 74(LS) with 74HC logic. The 74HCT244 can convert eight 74(LS) or 3V logic levels to 74HC 5V logic levels.
The 74HCT has a operating voltage range of 4.5V to 5.5V and the 74HC 2V to 6V.
The 74(LS)07 can't always be substituted with the 74HCT07, because the 74(LS) version's output can be connected to 15V via a pull-up resistor, but the 74HCT07 has a diode connected between its output and VDD, so it can't be connected to higher voltages, than its supply. In this case it doesn't matter, since the output is connected to 3V.