Just to add a quick note to the excellent answer above, the poems you mention are similar in that they both deal with the same or similar situations: WWI. But they contrast in that Lovelace's poem presents the common pre-WWI view of war, and Owen's presents the common view of war during and after the war. I say "same or similar" situations, because Lovelace never saw action. He died before he made it to the front lines. Owen, however, died in action or as a result of it. This difference in their situations is demonstrated by their different views of battle.
War had been presented as glorious and honorable in literature for centuries. Lovelace's poem reflects this. Owen's poem takes a turn toward the modern in literature. The illusion of the grandeur of fighting for one's country is stripped away in Owen. His works, including this one, strip away illusions.
In America, the view of war as glorious was partly exposed as illusion as a result of the Civil War. In England, however, at least in literature and popular culture, WWI caused this.
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.
I'm a retired soldier and a long student of history. From my perspective both teachers, while providing good information, missed a couple of points.
First, Lovelace was a 17th Century aristocrat with ties to the English royalty. In that environment a man's word was, indeed, his bond. If he had promised to support his King and then failed to do so he would have been publicly disgraced and probably had to forfeit all of his holdings and wealth and gone into exile. No respectable woman could have anything to do with him. In large part that attitude remained until the turn of the 20th Century; in the US there are many accounts of soldiers in the Civil War who only went to war because thier "honor" demanded it.
By contrast, the war that began in 1914 with great partiotic fervor by 1917 was a horribly bloody stalemate. In the UK and in France the cannon fodder enlisted men, and many of the junior officers, knew that the odds were very high that they would suffer or die for nothing at all. Owen, Sassoon and others wrote similar poetry based on their experiences.
I don't know when Lovelace wrote "On Going To The Wars." It would be interesting to see if he had actually engaged in combat before writing it.