Worlds adopted from Greek and Latin since the Renaissance have greatly enriched English vocabulary and especially increased the number of its synonyms, but this influence from the classical languages may be regarded as something between a help and a hindrance comment.
jk180 contributes some great information. I disagree, however, with the idea that we could get rid of all Latin- and Greek-influenced vocabulary (including the word 'vocabulary'!) and substitute native Germanic-based words. In English, there are a lot of native words with lassical counterparts that have stayed in use because the native and classical words developed slightly different meanings, or connotations. For example, take the words baby and infant. Baby is from the Middle English baban, and infant is from the Latin īnfāns. We use both words, and we use them in slightly different ways. Baby is somewhat informal, while infant is often used for official, formal, or medical contexts. I've never heard a mother say "my infant looks so precious when he's sleeping!" -- but we seem more comfortable discussing the "infant mortality rate" than we would the "baby mortality rate". Somehow the first one, because of its classical formality, provides the necessary distance for discussing clinical or governmental issues (infant is also the word of choice for the formal US Census).
The article linked below, on the etymology of food words, provides some helpful background information for the etymology of English vocabulary more generally.
pohnpei397’s answer is right on the money, I think. The huge influx of new words, mostly from Latin and Greek, in the Early Modern period enriched the English language immensely, but this influx also vastly increased the level of difficulty of the language for everyday readers. It’s not at all a coincidence that, in the Early Modern period, dictionaries were being printed and sold on a wide scale for the first time in the history of the English language. Simply put, these new words were often not familiar to general readers, and these readers were probably eager to purchase books that might help them become more familiar with all the new words.
Robert Cawdrey's Table Alphabeticall (1604) is often regarded as the first English language dictionary. The very elaborate title page to this dictionary promises that this volume will make many of the strange, new words accessible to people who haven’t dedicated their lives to learning. The book promises to present [with my brief comments added in brackets]:
the true writing, and vnderstanding of hard vsuall English wordes, borrowed from the Hebrew, Greeke, Latine, or French. &c. [In other words, the book will give correct spellings and definitions...]
With the interpretation thereof by plaine English words, gathered for the benefit & helpe of Ladies, Gentlewomen, or any other vnskilfull persons. [...in everyday English for people who aren't all that educated...]
Whereby they may the more easilie and better vnderstand many hard English wordes, which they shall heare or read in Scriptures, Sermons, or elswhere, and also be made able to vse the same aptly themselues. [...so that they might be able to understand these difficult words and even use them correctly themselves.]
The full text of this dictionary can be found here: http://www.library.utoronto.ca/utel/ret/cawdrey/cawdrey0.html
An image of the title page can be found here: http://www.bl.uk/learning/images/texts/dict/large1321.html
The only thing I can think of to argue that this wasn't a benefit is that it gives us a whole lot of new vocabulary (in a foreign language) to learn.
For example, if we just called the telephone a "long talker" or "long distance talker" we would have the same idea without having to learn that "tele" means "far" and "phone" means speech or talking. This would be a lot easier.