Oops, here's a cue for me to jump in and I almost missed it!

Ooof, it happens quite often that there are differences in the meaning of the same words in Hispanic American Spanishes and the Spanish spoken in Spain. Sometimes these differences are quite extreme, more often they are just differences in secondary meanings, or they stem from environmental differences (e.g. the word "plátano" refers to different versions of the same fruit in Latin America and Spain.)

Some of the more extreme ones that spring to mind just now:
pollera : in Spain, a woman that sells chickens; in many Lat. Am. countries it's a skirt.
cuadra : in Spain, a stable; in some Lat. Am. Spanishes it's a block, like a block of houses.
saco : in Spain, it's a sack; in some Lat. Am. Spanishes it's the jacket belonging to a suit.
de pronto : an adverbial which in Spain means "suddenly", whereas in Lat. Am. it means something like "maybe".
Finally, a well-known one is coger , which in Spain means "to take" or "to grab", while in many Lat. Am. countries it means "to f***". Mind how you announce that you're going to get the bus, as it might get you arrested instead...

What I'm not so familiar with is differences between the Hispanic American countries themselves, though I'm sure there must be quite a few too.