The sea is a useful symbol in the poem because it can be used to contrast the awesome power of nature with the speaker's emotional weakness and vulnerability. The thoughts that rise within him are like waves, but unlike waves, they cannot manifest themselves—much to the speaker's deep regret.
The speaker cannot articulate his feelings, as he is unable to move on from his grief; he cannot be strong himself, so he urges the sea to show him the way. He thus commands the sea to "break, break, break"—to crash down upon the rocks with full force to comfort him with a stirring example of nature's strength.
Yet in this case, not even the mighty sea, with all its power, can offer the sorrowful speaker the slightest crumb of comfort. Everyone else associated with the sea, whether it's the sailor lad or the fisherman's boy, can go about their lives, but not the speaker. The sailor, the boy, and the sea itself will all go on as they must, but the speaker cannot. Destined to remain forever paralyzed by grief, he is in the exact same position at the end of the poem as he was at the beginning.