The bearing of our griefs and carrying of our sorrows very clearly presage the atoning mission of the first coming of the Messiah. In that appearance, he comes not to rule the world, but to redeem the individual. Isaiah continues the them of rejection, highlighting the sad incongruity of Israel's rejection of the very one who had come to lift their burdens from them. "The word "sorrows" is more literally "pains," and the word "grief" is more literally "sickness." (Barney, Kevin. "Translation question on Isaiah 53:3" January 16, 2000, Scripture-L.)
As Barney also suggests, the reversal of the two elements "sorrows/pains" and "grief/sickness" creates a tighter poetic coupling between verses 3 and 4, with a chiastic reversal of the elements from one verse to the next. This suggests that there is a poetic point that is being made, and is probably made on the "pains/sickness" meanings. The atoning Messiah is a man of pain and sickness because he is mortal, and his undertanding of our pain and sickness allows him to carry those burdens for us.
It is possible that the somewhat cleaner connotations of griefs and sorrows grew from the tendency in Christianity to emphasize Christ's perfection, which of course (in that line of thought) would not allow such a thing as a physical ailment such as a sickness, nor perhaps even the mortal pain associated with manual labor. This later tendency to remove Jesus' essential humanity in favor of exalting his obvious deity does not describe the Messiah Isaiah, who was human, and subject to physical ailment and labor. It is that contradictory image of the very human and the very divine that is at the heart of Isaiah's declaration.