Why is the voltage gain in the simulator so different from the expected gain?
I'm seeing a gain of about 330 when I simulate that circuit with the default 2N3904 model at 1 kHz. Are you sure you're not dividing by two somewhere?
This is a bad circuit design. Here are some problems I noticed:

You do not give an AC signal frequency. The frequency is also missing from your capacitor calculation. If the frequency is low enough (e.g. 1 kHz), the capacitor could be reducing your gain. You want the capacitor's impedance to be much lower than \$R_2\$ or \$r_e\$.

Your calculation is heavily dependent on beta, which is not a reliable parameter. For example, the default 2N3904 model specifies a beta of 140. Your \$I_C/I_B\$ is not 140 because the simulator uses a more complex and accurate model of the transistor's behavior.

Your DC biasing circuit is unreliable -- there is no emitter resistor, and the base voltage is set with one resistor instead of a voltage divider. The biasing circuit from the second half of your video is much better.

25 mV is not a good approximation for \$V_T\$. Use 25.9 mV instead.


Going with your \$I_C\$ and \$V_T = 25.9 \mathrm{mV}\$, I get \$r'_e = 0.547 \Omega\$, which suggests a gain of 457. But again, this is a horrible circuit, so the gain being off by ~30% isn't surprising.
Add a 10-ohm emitter resistor and do the calculations again. You'll find that the gain is much closer to what you'd expect.
Leaving out the emitter resistor makes the gain larger at the cost of being less controlled. Real-world amplifiers like op amps use negative feedback to turn a large but uncontrolled gain into a small but precise gain.
EDIT: You should choose the capacitor so that its 3dB point is much less than the signal frequency. If you use the \$1/2\pi f R_{in}\$ formula at 1 kHz, the capacitor will drop ~30% of the input voltage.
Here's a graph showing the voltage division between an 80-ohm resistor and three different capacitors. At 1 kHz, both the 1.5 uF and 4.0 uF capacitors have significant voltage drop. It's best to pick something larger, like 100 uF. You can do AC frequency analysis in your simulator to see the effect on the gain of the amplifier.
What transistor are you using in your simulation? You could try a few different ones.
The Early effect reduces gain, but I don't think it's by that much.