To what extent is modern day English the same language as that introduced to the British Isles one and a half millenia ago?
All modern languages change with time, and English is no exception. Languages which do not change are normally "dead" languages, a classic example being Latin (no pun intended.) Anyone who reads Shakespeare will notice immediately a difference in the spelling of certain words; in fact those words were pronounced more nearly as they are spelled than they are now. The English of which you speak, commonly known as Old English, is so far removed from Modern English that modern English speakers would not recognize it as English. It is much more similar to German and the Scandinavian languages than to modern English. The following is an excerpt from Beowulf in the original Old English.
Hwæt! We Gardena in geardagum,
þeodcyninga, þrym gefrunon,
hu ða æþelingas ellen fremedon.
Oft Scyld Scefing sceaþena þreatum,
monegum mægþum, meodosetla ofteah,
egsode eorlas. Syððan ærest wearð
feasceaft funden, he þæs frofre gebad,
weox under wolcnum, weorðmyndum þah,
oðþæt him æghwylc þara ymbsittendra
In more modern English, it would read:
Listen! praise of the prowess of people-kings
of spear-armed Danes, in days long sped,
we have heard, and what honor the athelings won!
Oft Scyld the Scefing from squadroned foes,
from many a tribe, the mead-bench tore,
awing the earls. Since erst he lay
friendless, a foundling, fate repaid him:
Even some of the lettering is unrecognizable. The first word (Hwaet!) can loosely be translated as "Hark!" an old form for "Listen!"
It seems impossible that modern English would have developed from this language, but it did. Aside from changes caused by time, the language was also affected by the introduction of Norman French and Danish after the Danish invasion of the tenth century. So Modern English derives from Old English with some additions and modifications along the way; but bears little resemblance.