So we got a laptop at work with a piece of equipment, but it only has 8GB of RAM I want to upgrade the ram but old sticks I have won't fit.

(What's the difference??)

Anyways, should I just buy another just like this and take it to 16 GB or should I buy something different?

If you can find another stick of the exact model size and specs you go that route. Or just buy a new matched pair (most places sell 16gb as 2x8gb in a package) that meet are the same specs of what you have.

for $20 I bought one off amazon, from a highly rated seller that has the exact same sticker.

8GB, 1Rx8, PC$ - 3200AA-SA1-11

Even has the same M471A1K43EB1-CWE.

Now just to figure out if the graphics card has to be changed/updated.

Yeah Laptops arent too modular like Desktops. Generally speaking the CPU and GPU are soldered onto the MOBO

I haven't been able to find a good video of taking this thing apart, so I paused with 20 screws out of the back cover...

That said, some of these "tough" book type laptops apparently have graphics cards that are connected via some kind of posts with minimal solder. I got the RAM and installed it, it's been working nice so far, but will see how it handles the work load next week.