A good analogy is that the difference is like those in British and American English, but what are those differences exactly? Is Spanish in Latin America a branch from that in Spain?
The analogy is essentially correct, except:
- There are no spelling differences: only vocabulary, grammar and pronunciation
- There is a central body for Spanish (Real Academia Española), which keeps the "official" dictionaries, covering the usage of Spanish in all the world
One thing to consider is that there are also lots of differences between different latin american countries (Spanish in Mexico vs. Uruguay), and even inside big countries (Buenos Aires vs. Córdoba), much like the differences you'd find between England vs. Australia or Texas vs. New York.
Now, what are some examples of those differences?
Vocabulary: peach is called melocotón in Spain, durazno in Latin America
Pronunciation: In Spain z is pronounced approximately like the th in thick. In Latin America, it's pronounced like s (which has the same sound as in English)
Grammar: in particular, the second person has completely different forms. For some details, read the wikipedia article about Voseo.
The greatest differences are in the use of vosotros and in the pronunciation of
z and the soft
In Latin America, the informal 2nd person plural form vosotros and all of its verb conjugations are omitted in favor of the formal 2nd person plural Uds and its verb conjugations (which match conjugations of the 3rd person plural).
Where in Latin America the soft
c (in vencer) and the
z (in concozco) make a sound equivalent to the 's', in Spain, they sound closer to our English
th (as in thin)
There are lots of differences, and that's ignoring the variations inside Latin America. However, there are no spelling distinctions.
The most important differences lie the pronunciation of s, c (before e or i) and z. In Spain, z and c are pronounced /θ/ (like English th in think), while s is an alveolar /s/ (like English s, but slightly closer to sh /ʃ/). In Latin American, all three are pronounced /s/, and it's a slightly more dental s (just a bit closer to English th). This is called seseo, and it's also common in some parts of Spain.
A native Spaniard can correct me on this, but I believe ll and y (before a vowel) can be pronounced /ʎ/ and /j/ respectively (the first is a palatal version of l, the second sounds sort of like and English y), while in Latin America they are pronounced the same and can range from /ʝ/ (palatal fricative) to /j/ (again, like English y) to /ʃ/ in places like Argentina and Uruguay. This pronunciation is called yeísmo.
Again, someone while lives in Spain can tell me if this is right: the verb and pronoun forms for the second person plural are also different. While in Spanish the conjugations are vosotros amáis, vosotros teméis, vosotros partís, in Latin America it's ustedes aman, ustedes temen, ustedes parten, like the third person plural.
Aside from those, there are very many vocabulary and idiomatic distinctions, too many to list here. Wikipedia has a list of Spanish dialects. Also, remember that there are a lot of variations, both inside Spain and in America. On of the most notable, for example, is voseo: the replacement, in Argentina and Uruguay, of the seoncd person singular pronoun tú for vos and its verbal forms.