Is there a difference between TI brand of the 74HC595 and the NXP brand?
When it comes to these sorts of logic ICs, if you match the part number you'll match functionality. So, a 74HC595 from NXP should 'work' the same way as a 74HC595 from TI. The pins are given different nomenclature by the manufacturers, but they do map 1:1 between the ICs.

If there were a functionality difference, then there would be a corresponding change to the part number:


Consider the difference between a 74HC595 and a 74F595 - both do the same logic function, but the underlying technology of the IC is difference (HC is high-speed CMOS, F usually refers to Fast which generally implies Schottky construction). There can be differences in timing and logic voltage levels as you change technology.
Consider the difference between a 74HC595 and a 74HC596: the 595 is what you're using (8-bit shift register with output latches, three-state parallel outputs) whereas a 596 is slightly different (8-bit shift register with output latches, open-collector parallel outputs). 


Of course, even with the same number, there likely is very minor variations between them.  The fundamental functionality and speed will be compatible and unless you're really pushing the limits of the IC, you should be able to substitute with confidence. (That being said, always test!)

The timing diagrams should clear things up: