My favourite part of the whole sessions series at ~23:35, daddies happy smile.
I can see several discrepancies, your color ramp controlling the alpha is mostly white (No Alpha). The emission strength is at 20, I think Kent settles on 0.2, you have 3 cameras in the outliner, which one are you rendering?
You’re disk is in front of the lights and you have set shadow mode to alpha hashed.
I don’t know which or if any of these are causing the issue, but there are many potential problems.
I think you should re-adjust the settings to closer match the video then adjust to your liking from there.
As mentioned, if this is still a problem, share the blend file and we can better track down the issue.