There are various differences between the Spanish in Spain and Latin America but they are minor and it is certainly not difficult for someone from Spain to understand someone from Latin America. Any differences are similar to those, for example, between American and British English or Australian and Canadian English.
The Spanish spoken in Spain is known as Castellano and the Spanish spoken in South America, North America, Central America and the Caribbean is known as Español Latino. The Spanish language is said to have originated in the Spanish province of Castille and that is why it is known as Castellano (Castillian). Columbus took Spanish to the Americas in the 15th Century and, as Catholicism grew in The New World, Spanish became the main form of communication there.
A well known grammatical difference between the Spanish in Spain and Latin America is that Spain uses two forms of the plural ‘You’: Vosotros (informal) and Ustedes (formal), but in Latin America people only use Ustedes and never Vosotros. Tú (informal) and Usted (formal) are used for the single ‘You’ in Spain. You will hear Tú being used in Latin America but it is much less common than Usted and in some countries such as Argentina and Uruguay Tú is replaced by Vos.
Another grammatical difference is in the way that people in Latin America tend to use the Past Simple tense when describing all actions that have been completed, unlike in Spain where people will use the Present Perfect tense to describe completed actions that ‘have happened’ the same day. In Spain, for example, people would say “Esta mañana hemos ido al parque” (This morning we have gone to the park) whereas in Latin America people would say “Esta mañana fuimos al parque” (This morning we went to the park).
You will notice pronunciation differences between the Spanish in Spain and Latin America. People in Latin America pronounce the ‘C’ (before ‘I’ and ‘E’) and ‘Z’ as ‘S’, rather than the Castellano ‘TH’ sound. Gracias, for example, will be pronounced GraTHias in Spain and GraSias in Latin America. This is because the majority of Spanish emigrants to Latin America came from the South of Spain where people also pronounce ‘C’ and ‘Z’ as ‘S’. Another pronunciation difference can be noticed in Argentina and Uruguay where the ‘LL’ and ‘Y’ are pronounced as ‘SH’, rather than the Castellano ‘Y’ sound.
Finally, as with American English and British English, there are many instances when vocabulary differs. For example, the word Coche is used in Spain for ‘Car’, whereas in Latin America you will hear Carro or Auto. Conducir is the verb for ‘Drive’ in Spain and in Latin America it is Manejar. ‘Mobile phone’ is Móvil in Spain and Celular in Latin America. ‘Banana’ is Plátano in Spain and Banana in Latin America. ‘Boat’ is Barco in Spain and Bote in Latin America. And ‘Potato’ is Patata in Spain and Papa in Latin America.