These two poems convey a very different view of war. In "To Lucasta" the poet takes on the persona of a lover speaking to his beloved, explaining why he must leave her to go to war to "chase a new mistress." He states that he must do this to be honorable for it is his honor that enables him to love her so much. The tone is idealistic and patriotic.
I could not love thee, Dear, so much
Loved I not Honour more
The imagery is that of a gung-ho young man, going off to war - a hero, perhaps.
Contrast this tone with that of "Dulce et Decorum Est":
Bent double, like old beggars under sacks,
Knock-kneed, coughing like hags, we cursed through sludge
This presents a very different view of war -- the actual fighting. This poem describes a battle and a death in the battle. Look at the strong words that are used to depict the action: haunting, blood-shod, drunk with fatigue. This is what war is really like, the poet implies. So, don't give me that lie that it is glorious to die for one's country (which is the translation of the title - Dulce et decorum est pro patria mori).
My friend, you would not tell with such high zest
To children ardent for some desperate glory,
The old Lie; Dulce et Decorum est Pro patria mori
Owen is considered a major poet of WWI. You can read about him here on eNotes.