The repetition of the word "break" suggests that the waves keep breaking, over and over again. It suggests that things change and time moves on ceaselessly: regardless of human actions. The waves crash on "cold gray stones," a phrase (especially given the grieving tone of the poem) that suggests a melancholy and a comparison between the stones and grave stones.
The waves (nature) crash and wash over the (grave) stones. The speaker is so overcome with emotion that he can not find words to match his thoughts. The waves washing over the stones suggests that nature (and time) do not pause to mourn the dead (as the speaker does). The waves are wild, free from grief and care. In this stanza, the waves and the sea represent nature and the unceasing movement of time. From the speaker's perspective, the sea means these things, but also the notion that nature (personified) does not care to note the speaker's grief.
In the second stanza, the children play freely and without care. This just shows a comparison of their joy and the speaker's grief. But the children's carefree spirit could be paralleled to the allegory that nature is also carefree; that is, nature and the children are not affected by the speaker's grief. The children are still young and wild (like the sea); not yet burdened with life's hardships. The waves will always break, children will always play, time will always move on.
The two stanzas differ in emotional tone. The first is quite melancholy. The second gives illustrations of joyous play, but the speaker does note these illustrations with a kind of sarcasm and/or mocking tone: "O, well for" them who don't have to deal with such grief.