This variation is akin to that of the differing varieties of English; an American would have no trouble identifying an Englishman based on his London cockney or an Irishman based on his brogue. The same is true for Spanish; a native of Madrid would immediately notice a difference in the way her friend from Guadalajara or her cousin in Paraguay speaks.
Of course, just like with English, native speakers from all over the Spanish-speaking world are able to understand each other, even though they may have a number of slightly different lexicons and some variations in grammar. Thus, what is called “Spanish” today is in fact a collection of different accents of the same mutually-intelligible language, each with its own history and characteristics.