There are considerable grammatical differences. Hungarian is an agglutinative language: what words do and mean in a sentence is affected by the bits that are added on, before, after and within them. So 'the table' is 'az asztal' and 'at the table' is az asztalnál'; 'me' is 'engem', 'by me' is 'nalám', and 'my flat' is 'lakásom', with the vowel plus -m providing the 'me' component.
All prepositions are added on to the end of the words they relate to, word order is the same for questions and statements, and there is not a masculine/feminine gender distinction.

Origins

The modern Hungarian language, with its eight major dialects, is a descendant of the ancient Uralic language family which is thought to have emerged many thousands of years ago in the area of the northern Urals in Russia. Like its distant cousin, Finnish, it belongs to the Finno-Ugric branch which numbers some 15 or so related languages scattered across Eurasia.

In its early forms it was spoken by members of nomadic tribes from the steppes. These spread southwest in a series of migrations before settling in the location of what is now modern Hungary, in the Carpathian basin, the great plains through which the rivers Theiss (Tisza) and the middle Danube run. This settlement was probably completed towards the end of the 9th century AD.
The name 'Hungarian' is derived from on-ogur, the 'ten tribes', which was the name of a tribal confederation formed during an earlier stage of the migration. The Hungarian name for themselves is Magyars.

Pronunciation of Hungarian

Consonants sound more or less as they do in English, though there are some exceptions, in particular the so-called consonant clusters, cs, gy, sz and zs. These function as separate letters and usually do not occur at the beginning of a word. They are pronounced as follows:

cs as the 'ch' in 'church' - e.g. csütörtök -Thursday ( Days of the Week)

gy as the 'j' in 'jury', but lighter. Listen to gyógyszertár - chemist (Places and Buidings 1)

sz as the 's' in 'salt' - e.g. szerda - Wednesday and szombat - Saturday (Days of the week) 

zs as the 's' in 'pleasure - e.g. rizs - rice (Words to do with food. General)

Other different Hungarian consonants:

c as the 'ts' in 'hats'
j as the 'y' in 'yes'
ly more of a 'y' than an 'l' sound. Listen to hüvelykujj - thumb in Parts of the Body
ny as in the Russian nyet
r trilled as in Scottish
s as 'sh' in 'shop'
ty as the 'tu' in 'tube' or 'tune' in British English
w only occurs in imported words, and is pronounced as 'v'

When consonants are doubled, sound both elements, so 'butter' would be pronounced 'but-ter'. When consonant clusters are doubled, only the first letter is affected, e.g. a double gy becomes ggy.

Vowels are less straightforward. In general vowels can be related to their English equivalents by remembering that an accent lengthens them. It is important to remember that an accent not only alters the sound, it can also alter the meaning.

A good section to visit is Countries and Where I'm From, in which 9 of the 14 vowels listed below can be heard. Directions 2 conveniently provides the full range of unaccented vowels.

a tends towards the 'u'in 'hut', 'but'
á sounds like 'a' in 'father'. Compare the two 'a's' in kanál - 'spoon' (Eating)
e short English 'e' as in 'get', 'bet', etc