Why do written English vowels differ from other Latin-based orthographies?

Other interesting references on the Great Vowel Shift:

• See and Hear the GVS (excellent visual and audio!)
• Brief History of English
• More English History

The Great Vowel Shift would probably be just an historical curiosity if it weren't for the fact that the first printing press opened in London in 1476, right in the middle of the shift!

Before the printing press was invented, the words in handwritten texts had been spelled according to the dialect of the scribe who wrote them. However, book production was slow and few people could read in any case.
The early printers used the older spellings which Middle English scribes had used. They didn't understand the significance of the pronunciation changes that had just gotten well underway.
By the time the vowel shift was complete (about 100 years from start to finish), hundreds of books had been printed with the older spellings.
The new high volume of book production combined with increasing literacy proved to be powerful forces against spelling change.
As a consequence, many spellings have become "fixed" to the Middle English pronunciation, rather than the modern ones, and we still spell the word for the earth's satellite as "moon."

There may be some truth in the 'vowel shift', but if u study texts with original spellings from 1350 to 1750, it is very difficult to find much evidence for it, because between 1430 and 1650 English spelling became increasingly varied and random. Between 1525, the publication of the first English New Testament, and 1611, the year of the King James Bible, most English words acquired several spellings. This can be seen in the writings of Sir Thomas Moore and the highly educated Elizabeth I.

What the study of old texts reveals very clearly is that English spelling was repeatedly deliberately messed up, leaving the relationships between sounds letters increasingly random. The first time was over 1000 years ago, when some monastic scribes disliked having to write sequences of short strokes next to each, as u would get with a sensible spelling of 'munth'. So they substituted o for u next to n and n (month, front) and also next to v which back then spelt v and u (hence the name double u for w): love, glove, wonder.

Around 1430 Chancery Clerks were suddenly obliged to switch from French to English. They took their anger about this out on English spelling. They severely undermined Chaucer's consistent use of e and e-e (bed, bred, erly; seke, speke, reson), with irregularities which persist to this day. They not only added ea to the English alphabet but used it for several sounds as well (treat, great, threat).

Early printers messed up English spelling a bit more, by adding extra letters to earn more money, as they were paid by the line. Many of them also spoke no English and made all kind of random changes (e.g. sadnes to sadnesse, frend - friend, bild - build).

The final blow was dealt by Samuel Johnson who wanted to force English into a Latin mould. He was most responsible for undermining the English use of doubled consonants for marking short vowels (rabbit, merry, poppy) by exempting many Latinate words from the system (habit, very, copy) or using doubled letters to show defunct Latin prefixes (e.g. adplicare - apply).

For a fuller explanation of the above see http://englishspellingproblems.blogspot.co.uk/2012/12/history-of-english-spelling.html

The previous answers are all right on the money but there is an aspect of this that seems to never be discussed. The reality with English is that Britain had a lot of overlapping linguistic and cultural influences from the High Middle Ages through the Early Modern Period, moreso than most cultures. If you look at English during its earlier phases there was a substantial effort to keep the orthography sensible and consistent with pronunciation. The Norman Conquest, which came somewhat close to wiping out English, substantially changed the language and introduced French orthography, which itself is complicated, though still at least mostly consistent. With the lingering effects of the Renaissance and the desire of later English writers to imitate Latin spelling more the connection between writing and pronunciation simply snapped. The pronunciation was changing due to the Great Vowel Shift and yet writers were more interested in mimicking French and Latin spellings than following their own pronunciations. The spellings that were finally arrived at made no sense from an English perspective though they make some sense if you know French, Latin, and some things about Middle English. Certainly similar pressures have existed in other language groups, but not to the same degree as in Britain. In essence you could say the British stopped trying whereas most cultures have continued to make some effort to make their orthography phonetic.