The average modern English speaker would have trouble understanding Shakespeare, which is quite unfortunate.
In terms of Latin to Modern Spanish, I do think they are both vastly different. Sure, there are many cognates, but Latin grammar is way more complex than Spanish grammar (the same applies to other Romance languages such as French, Italian, and Portuguese, which I also have studied). While I don't know Old English very well, I do think it is more similar to Modern Standard German (probably more similar to Old High German, actually, because of its inflectional system) than Modern English.
Ancient Languages are so important. I have learned a lot more about English (and other modern languages) through the study of Classical Latin, Ancient Greek, and Old Icelandic. We do need to value them so so much...specially to avoid having young, eager freshmen tell me "So Shakespeare is like Old English, right?"