The differences between European and Latin American Spanish came about simply due to distance and time. As the Spanish began to set out and colonize the west, cultures collided and certain expressions, words, and phrases began to mix with other linguistic influences. Over time, this evolved to create permanent changes in the way the language is expressed, both in speech and in writing, creating distinct Spanish variations.
The indigenous people of Latin America exerted a great influence on the Spanish language, and as a result, differences between the two types of Spanish began to emerge. This was mainly due to the diminishing contact between Latin America and Spain, coupled with the increase exposure to languages used by the inhabitants of the Americas.
Other languages and neighboring countries have also affected the Spanish language. American English from the north continues to influence Latin American Spanish vocabulary to this day. One notable trend is the higher abundance of loan words taken from English in Latin America, as well as words derived from English with the foreign spelling sometimes left intact. For example, the Latin American Spanish word for "computer" is la computadora (as opposed to the Spanish word ordenador, which is much more similar to the neighboring French word ordinateur).
You will find that many differences in vocabulary reflect features unique to nature, family culture, food, drink, society and politics. New words emerge as needed in some areas, but the same concepts and corresponding words may not exist in other places. In Spain, lemons and limes are both very common, and are known as limones and limas. In countries where lemons are uncommon – like much of Latin America – the word limón may be used just for limes (or even for both), as they rarely have to differentiate between the two.
Here are some examples of words commonly used in Spain, and how they might be translated for a Latin American audience. Watch out for those that may have other meanings!