Can you explain me what are the difference between inverted and non-inverted mode?
It's rather self explanatory, but here's a short explanation. Inverted serial inverts each bit. I.E. if you send a 1, it's a 0. If you send a 0, it's a 1. That's what inverted means.

The PIC pin that's sending serial data, in inverted mode, will idle (fancy name for when it's not sending serial data) at logic 0.

Standard serial data (without a parity bit) consists of 10 bits. 1 start bit, 8 data bits, and 1 stop bit.

The start bit, with inverted mode, will be a logic 1. Then it sends your 8 bits of data, and ends with a stop bit. The stop bit is going to be the same as the idle logic. I.E. for inverted serial mode, the stop bit will be 1 bit time at the idle logic state of 0.

TRUE or "non inverted mode" is just the opposite logic levels.

When must I use each one?
That totally depends on the external device you're communicating with serially, and it's something you "as the designer" will need to determine.

The MAX232 is an inverter. The PC serial port expects an idle logic of 0 (this is actually -10 or more).

So, if you're going to communicate with the PC serial port, using a MAX232 RS232 converter, with a bit-banged serial command like SEROUT, you'll want to use TRUE mode with your PIC since the "idle logic" with TRUE mode is logic 1. 1 on the input of the MAX232 = 0 on the output of the MAX232 connected to the PC serial port, and that's the logic it expects during idle periods.

This is "inverted" through the MAX232, and places a -10 "or more" on the PC RX pin during idle periods.

If you had a direct connection from your PIC to the PC RX pin, like shown in your PBP manual, then you would of course use inverted to hold the PIC TX pin at logic 0 during idle periods.

In short, it's up to Y-O-U as the designer to know which mode to use based on the device you're attempting to communicate with.

So Mister_E answer is wrong
No. He was right. Maybe you just didn't understand what he was saying...;o}