The central theme of “Ulysses” is that there is a search for adventure, experience and meaning which makes life worth living. Tennyson used Ulysses as the old adventurer, unwilling to accept the settling of old age, longing for one more quest. Tennyson also wrote this in memory of his friend Arthur Hallam.
In the second section, Ulysses implies that Telemachus is better fitted to be a king, which requires more governance, discipline and patience. And he (Ulysses) is better fitted to be an adventurer. This is what he means when he says, “He works his work, I mine.”
One might think that he is being selfish, wishing to abandon his wife in exchange for one more adventure; and this criticism is justified. However, for Ulysses, life is meaningful only if there is some quest. In the first section, he says,
How dull it is to pause, to make an end.
To rust unburnished, not to shine in use! (22-23)
He is saying how boring it is to rest at home; that he essentially rusts from lack of use.
The last section is a speech intended for his men. However, if this is the crew of the adventures in The Odyssey (Odysseus is the Greek name for Ulysses), it must be understood as a day dream Ulysses is having or he is contemplating one last rousing of the troops for another quest despite their, and his, old age:
We are not now that strength which in old days
Moved earth and heaven, that which we are, we are--
One equal temper of heroic hearts,
Made weak by time and fate, but strong in will,
To strive, to seek, to find, and not to yield. (66-70)
Their will to seek adventure will overcome what limits age has brought upon them. Ulysses is not satisfied to sit and contemplate on his past adventures. He is somewhat egotistical but that egotism is secondary to his deep desire for adventure. In other words, he is proud of his reputation, but resting on those laurels is meaningless. This refers to that line where he compares himself to a tool, presumably a sword, that only shines if it is in use.