Why does the analysis consider Ng5+ better than Ne5+?
Because white wants to get rid of the annoying knight on e4 and open the file for him, so Ng5+ Kg8, Nxe4 dxe4 , Qxe4 and white is in a better position than Ne5+ (But that doesn’t mean that white is better than black since you are down pieces). If instead Kg8, Nxg5 is played by black, then Bxg5+ and white is in a winning position.
Why would black play Kg8 when he could take the knight with his bishop, knight or h pawn?
Because after 14.Ng5+ and either 14...Bxg5 or Nxg5 White wins immediately with 15.Bxg5+
After 14...hxg5, it looks like draw is likely.
The analysis assumes so because the cloud engine is poorly working.
I agree with the above comment Ng5+ being more logical, as it removes the e4 knight, but all things considered, both moves are of equal value.

Because if Ng5+ Nxe4 Bc7+ wins the queen and if hxg5 Bc7+ wins the queen so the potential of discovered attacks and the removal of the e4 knight is probably why Ng5+ is considered better than Ne5+
Why would black play Kg8 when he could take the knight with his bishop, knight or h pawn?
I forgot to mention that after hxg5, white can play Bc7+ winning the queen.
Why would black play Kg8 when he could take the knight with his bishop, knight or h pawn?
Because after 14.Ng5+ and either 14...Bxg5 or Nxg5 White wins immediately with 15.Bxg5+
After 14...hxg5, it looks like draw is likely.
No, Bc7+ instead Bxg5+ is better to win the queen. It is not a draw, but winning the queen.
Yeah I just realized that it is an equal position. I also misunderstood when you said Bxg5 (I thought you meant in the 15 move). Still, Kg6 is definitely hard to spot unless the opponent is a titled player playing classical, or it someone who uses engine.
Yes! Those tactics all look pretty simple when you've got an engine! I'm sure I would miss a lot of them playig OTB
Yes! Those tactics all look pretty simple when you've got an engine! I'm sure I would miss a lot of them playig OTB
One may prefer white here (queen for 3 pieces, and a rather vulnerable Black king), until he realizes that Black is threatening Ba6 winning an exchange, and you can prevent it only with the anti-positional c2-c4.