The last two uses of dedicated in the Gettysburg Address are as follows:

It is for us the living, rather, to be dedicated here to the unfinished work which they who fought here have thus far so nobly advanced. It is rather for us to be here dedicated to the great task remaining before us ...

Dedicated here means determined to complete. Lincoln is saying that those in the North who have lived through the Battle of Gettysburg must have greater determination than ever to win the Civil War so that the sacrifice of the dead Union soldiers will not have been in vain or worthless.

Lincoln then uses the word devotion at the end of the address, stating,

from these honored dead we take increased devotion to that cause for which they gave the last full measure of devotion—that we here highly resolve that these dead shall not have died in vain—that this nation, under God, shall have a new birth of freedom—and that government of the people, by the people, for the people, shall not perish from the earth.

Dedicated and devotion mean largely the same thing in the context of this speech. Lincoln is saying at the end that because so many have died, the Union must have increased devotion or determination to win—in fact, the North must now be to dedicated to the uttermost, to the "last full measure," to keeping the nation together.

Devotion, however, has a religious connotation. It means more that just dedicated to a project, but carries the implication that the soul as well as the body is completely committed to the task at hand. Using devotion interchangeably with dedicated shows that Lincoln meant dedicated, in the last two uses, to mean that both body and spirit must be fully committed to winning the fight ahead. He is saying that this is war to be pursued with the relentless vigor and faith of a holy cause.