So, I did major (and some minor) characters. Quick summary (for my sake, mostly): The King, The Prince, The Queen, The Love Interest, The Uncle, The Advisor, The Best Friends. K.
Number two was "major events of the story", and number three was minor events or events that happen outside the story. Ok, I had these separate, but I'm going to go ahead and do my (more) detailed plot analysis all at once instead of in two parts.
a) The Prince is born and raised.
TLK
b) The Uncle kills The King.
TLK
c) The Prince runs away/goes insane and The Uncle marries The Queen.
TLK
d) The Prince sees the ghost of The King.
TLK and H
e) The Prince waffles about what to do.
TLK and H
f) The Prince orders a play mimicking the action prior to the events of the story to get a reaction out of The Uncle.
H
g) The Prince kills The Advisor accidentally.
H
h) The Prince is exiled.
H
i) The Love Interest goes mad with grief over the death of The Advisor (her father) and commits suicide.
H
j) The Prince returns.
H
k) The Prince learns of The Love Interest's death and attacks Laertes, proclaiming his love for the dead Love Interest. (Laertes is The Love Interest's brother and son of The Advisor).
H
l) The Uncle orchestrates a duel between Laertes and The Prince. During the duel, The Queen drinks from a poisoned cup and dies.
H
m) Laertes cuts The Prince with a poisoned sword tip, mortally wounding The Prince. Laertes is also cut with the poisoned blade and dies.
H
n) The Prince kills The Uncle.
TLK and H
o) The Prince dies.
H
p) The Prince and the Love Interest live happily ever after, ruling wisely and well over a restored kingdom.
TLK
Of a-o, only 3 of the 16 events occur during the action of both works. Three more occur during the action of TLK and before the action of H. So that's approximately six of sixteen important plot points that occur in both works. There's a pivotal character in H (Laertes) who does not have an equivalent in TLK. So far, they look remarkably different.
Maybe a good next step will be to find other works with King/Uncle/Queen/Prince dynamic going on, since that seems to be the strongest similarity.