and I think of each life as a flower (line 15)
Flowers are notably short lived. So part of what this line does is to stress the evanescence of life. It also might suggest a connection between that brevity and the beauty and vibrancy that flowers also represent. Do we think flowers are beautiful partly because they're only around for a little while?
each name a comfortable music in the mouth, tending, as all music does, toward silence, (lines 17-18)
We could see this as another subtle way of recognizing that the beauty and splendor in life is tied to the fact that it is temporary. This line recognizes the loveliness of each name (and, by extension, each thing), by comparing it to music, which is a form of art, and one that is pretty much universally loved and appreciated. Then it points out that all music tends toward silence. What does all life tend toward? You guessed it! Just as music is defined by the silence around it, life is defined by death.
When it's over […] (lines 21 and 24)
Here, toward the end of the poem, we see that mortality has been rephrased. We began with "when death comes," and now we have "when [life's] over." Having life replace death as the center of this phrase seems to mirror the movement of the poem – beginning with a contemplation of death, it moves to a contemplation and philosophy on how to live.