Purpose of Pullup Resistor for Active low pins in EEPROM
If I understand correctly, you ask why should we use a pull-up (or pull-down) instead of directly connecting the pin to the power supply (or GND).

Well, the thing is, if the pin you want to pull-up is not used by any other active circuit, you can do both. Either use a resistor or directly connect to the power supply.

This seems to be the case in your application. You just connect HOLD and WP pins to 5V and you don't want to drive them otherwise. So what you have done is correct, not necessary to use resistors.

But imagine what happens if you wanted your MCU to drive these signals, as it shown in the application note of Microchip. If you had connected these pins directly to 5V, as you did, when the MCU would need to drive them LOW there would be a short circuit, which is obviously not good! If you had the pull-ups instead, as it is advised, when the MCU would drive the pins LOW there would be no short circuit, only some current flowing from the 5V through the pull-up to ground.

And then, why need pull-up anyway? Well, that is to have a well defined state on these pins, even when the MCU has not yet booted (at startup) and its pins are most probably floating. This helps avoiding accidentally driving these pins to LOW during this time.