Hi Katie:
First of all, I'm sorry for my English. I apologize for this.
The main difference between the Spain and Latin American Spanish. Is that in Castilla and throughout the north of Spain, CE, CI, and Z before any vowel has a different sound than the S. The S, in this part of Spain, and much of the country, it maybe sounds very strong, for many Latin Americans.
Another difference is the conjugation of verbs. But within Latin America are also differences.
As for the differences between the Spanish of Spain and Latin American countries, apart from the phonetic differences, what I most notice is the "tune" or music of the talk. In this sense, it don't look anything at all the Spanish spoken in Mexico, Argentina, Colombia, Dominican or Cuba. I couldn't distinguish between Uruguay and Argentina, but with a Mexican I could. Among Bolivians and Ecuadorians, although I have spoken with many of them, I still can't distinguish.
Finally, the biggest differences I think there are between the Spanish spoken in different countries are the words of each country, that are a lot. But a Latin American person who lives in Spain, a month of being here, apart from his accent or "tune" of which I spoke before, He speaks the same Spanish than we do because he quickly adopts the most used words here.
I sincerely hope this helps, Katie.