For Housman, the greenery does much more than just look nice in a vase or give you a really wicked, itchy rash. A.E. uses shrubs and flowers to symbolize things like victory, affection, and respect for the dead. And they make super-cool hats, too.
Line 10: These "fields where glory does not stay" work literally and metaphorically. On the one hand, the fields are, well, fields. The flowers and plants that bloom in the field making it beautiful in the springtime don't last—the field's "glory" fades with the changing seasons. Metaphorically, the fields are the playing fields where athletes compete and achieve glory—glory that fades when inevitable defeat comes.
Lines 11-12: The "laurel" and the "rose" work in two ways as well, literally and symbolically. The laurel is the plant and it blossoms "early" and the flower "withers quicker than the rose." These plants also function symbolically: laurel represents victory, and roses represented respect for the dead in ancient times. These days we're more familiar with roses symbolizing love and affection, but Housman, being the scholar that he was, would have known about the respect-for-the-dead thing. The glory of victory (laurel) withers faster than respect for the dead (the rose). By dying before suffering defeat, the young athlete has locked down the public's eternal respect. Well played?