How do the Greek myths function in Tennyson's poems "The Lotos Eaters" and "Ulysses"? Why did he go back to Greek and Roman myths in the Victorian Age? Does being the poet laureate of the age has any importance on this issue? If so, why?

"The Lotos-Eaters" alludes to the encounter with the lotus eaters had by Odysseus and his crew in Homer's The Odyssey.  The speaker tells of the lotus eaters approaching Odysseus's ship and the way they seem so "melancholy"; further, anyone who partakes of the lotus fruit they offer no longer wishes to return home.  These crewmen think,

Dear is the memory of our wedded lives,And dear the last embraces of our wivesAnd their warm tears: all hath suffer'd change:For surely now our household hearths are cold,Our sons inherit us: our looks are strange:And we should come like ghosts to trouble joy.

The men who eat the lotus have many ways to justify their desire to remain there and never return home again.  Though they recall home fondly, they have been away so long now they think that they would be like strangers to their loved ones and that their homes would have changed so much in their long absence.  There are so many reasons to stay.  In the Land of the Lotus Eaters, they can rest peacefully, like the gods, and not worry over humankind anymore, neatly escaping the mariner's death at sea for surely "slumber is more sweet than toil."  They decide that they "will not wander more."  In The Odyssey, Odysseus sends a few of his men to investigate the Land of the Lotus Eaters, and the ones who eat the lotus do, indeed, say that they prefer to stay there and never return home.  He actually has to overpower them physically to get them back to the ship.  This poem alludes to that episode, filling in the part of those crew members who eat the lotus.  There is no mention of Odysseus and his monumental journey; rather, this story is theirs, and they want to stay—not go.
"Ulysses," on the other hand, is narrated by Odysseus, and he wishes to go, not stay.  He finishes the poem by saying his will is "To strive, to seek, to find, and not to yield," and he seems to assume this same goal is possessed by his crew as well.  He refers to them all when he says, "We are not now that strength which in old days / Moved earth and heaven, that which we are, we are."  He believes that he and they are of "equal temper" and in possession of "heroic hearts."  However, when comparing his assumptions of his men to their own words in the other poem, we see that Odysseus is wrong: he wants adventure and danger; they want safety and peace.  
Thus, the two poems use allusions to the ancient Greek tale of Odysseus in order to show two sides of the story: in The Odyssey itself, we do not hear from the crew that ate of the lotus as we do in the first poem.  In the second, Tennyson gives us a sense of how self-centered and unaware Odysseus really is: he does not realize that he has imposed his own will on his men, who would rather have rested peacefully.