As a Spanish (Spain) speaking person I can notice the differences between European and American Spanish. Is there also such a big difference between European and American English?
Vocabulary and Phoneticaly wise.
No, there is less difference between American English and European English than there is between American Spanish and European Spanish. The reason for this is that the English were about a century behind the Spanish in the colonization of America. The means there has been more time for things to drift apart. Going the other way, look how much closer to European English that Australian English is than the American English in the United States and Canada. That's because the antipodes were settled even later than America was.
It's as easy to find differences in vocabulary between England and America as it is between Spain and America. That's always going to happen in languages spoken over such a large area. The same can be said for pronunciation, where just as virtually all American Spanish speakers "don't know how to say" z's and ll's, many European English speakers "don't how how to say" their r's. So that's all a wash for vocabulary and pronunciation.
But what really stands out in the Spanish-speaking world is that the pronouns and the conjugations of the verbs are quite different. This is grammar not lexicon, so it's much more striking. That doesn't happen in any of the Englishes.
Most American Spanish speakers have no second-personal plural vosotros, vosotras pronouns and corresponding verb inflections like habláis and hablad. In American Spanish one uses the formal third-person ustedes forms instead, which is a completely different person. And some countries in America use vos for the second-person singular instead of tú, which brings its own verb forms like vos hablás in some countries
It's very hard to find anything in English that's so dramatically different transatlantically in terms of grammar as this is in Spanish.