Why does one method work but not the other?

I have a LEDStrip that I am fading through the different colors, however when I was using just the 12V 5A into the barrel jack, the arduino got uncomfortably hot but lights worked as intended.

SO, I got a seperate supply to power the Arduino as you can see here, but now the lights do not fade...

They only stay white(bright too). The code is uploaded onto the arduino too.

When I changed it to this, it works fine agian... Why is this? I dont want to put 12 Volts into the Vin because of the fear it might make the voltage regulator hot.

It appears you have no GND between the 12 supply to the Arduino GND in your first image.

A signal from the Arduino (in this case your PWM outputs) needs to have a return path to the Arduino.

Connect the ground of the Arduino to the ground of the 12V (e.g. on the breadboard) and you should be OK.

Awesome! I'm learning! Thank you so much guys

What the black thingy ?

looks like it say FET-N, but there are not notes if it really is an FET or if it is a transistor
regardless,it is good to put a resistor between the Arduino pin and the base of a transistor or gate of FET

if your Arduino is getting hot, then there is a path that is allowing excessive current to flow.