How to demonstrate the effect of baudrate in serial communication
The answer is pretty obvious - baud rates define data transmission speed. If you are using AT commands and typing them by hand, you won't see any difference, because the bottleneck is you. If you are transmitting larger amount of data, there will be difference. Try setting up a serial transceiver in loopback (connect RX and TX together) and then send 1kB of data over serial. At first do it at, say, 1200 baud and then 115200. You will notice the difference, because first will go in  1024/(1200/8) 6+ seconds, other will go way under a second.

Usually people try to go for as high baudrate as the system can tolerate without failing. Only when transmission errors start appearing you would start reverting to lower baud rate.

I have a system that has to transmit images (5MB) over serial line, even at max standard baud rate of 921k it is too slow (110kBps, whole image takes 50s to transmit). I have boosted the speed to non-standard 8Mbaud, thus reducing the transmit time tenfold, at the expense of extra hardware and code - have to have packet verification and error checking, hardware that supports these high baud rates, etc.