Brick and mortar chimneys exposed to water over time will deteriorate. Though bricks are more or less indestructible the mortar holding them together is prone to damage and eventual failure. Once the mortar between the bricks cracks and begins to fall out the bricks and structural integrity of the chimney are sure to follow.
These pictures show some typical long term water damage to the mortar between bricks on a above roof chimney. Someone noticed this problem before and attempted to fix the failing mortar with silicone caulking. This is not the correct repair technique - caulking is not going to do anything to stabilize the brick.
This chimney needs repair - the bricks need to be "re-pointed". Re-pointing is the process of removing the old mortar joints and replacing with fresh mortar. This is within the realm of ability of most homeowners but it is a tedious job and depending on location could pose some safety concerns. If this project doesn't appeal to you or you are not comfortable being on your roof, look for a masonry repair specialist to do the job.
The following is a link to a nice overview of repointing-brick on the This Old House website.
Brian Jovag, owner of Jovag Home Inspection.