Crown against cap, the confusion worth settling first
These two words are used interchangeably by almost everyone who is not in the trade, and getting a quote for the wrong one is common. The crown is masonry: a sloped slab covering the whole top of the chimney, with the flue tiles coming up through it. The cap is metal: a cover with a mesh screen sitting over the flue opening. If a quote says crown and the photograph shows a metal box, the two of you are talking about different components and different money.
What a crown is supposed to do
It covers the top of the masonry so rain does not land on the brick and the mortar, it slopes so the water runs off rather than sitting, and it overhangs the brick face by enough to throw the runoff clear, with a drip edge underneath so the water lets go instead of running back onto the wall. A crown that is flat and cracked does the opposite: it holds the water and delivers it into the wall it was built to protect.
Why crowns crack here
Frost, with water behind it. The United States National Park Service, writing about brick masonry, names water as the most destructive force on historic masonry surfaces and states that it wicks into the brick cores, so that in a northern climate where winter freezes are prevalent the brick deteriorates from freeze thaw action. A crown is the most exposed piece of masonry on a house. It takes every drop of rain that lands on the stack and freezes with whatever pools on it through a northern New Jersey winter. The second cause is construction: a crown poured directly against the flue tile with no expansion joint cracks when the tile expands with heat.
What a cracked crown does to the rest of the chimney
Water that gets past a crown goes into the space between the flue and the outer masonry, which is the one part of a chimney nothing can be seen inside. It saturates the brick from within and freezes there, and the faces spall from the back. That is why a chimney with a failed crown loses brick faces on the outside with no obvious reason on the surface, and the crown gets repaired before the brickwork does. Repointing under a broken crown is money spent on a wall that is still being soaked from above.
What is done at the same visit
A crown, a cap and the top few courses of masonry are all reached from the same position and all share the same water. Repairing one and leaving the others for a separate trip means paying for access twice. The report names what is done together and what is a separate job, and the estimate is written before anything is started.
When this shows up
Crown damage becomes visible in late winter and spring, after the freeze and thaw cycles have opened what the fall rain filled. That is the same window as spalled brick and white staining, so crown work, repointing and waterproofing sit in the spring calendar here and sweeping sits in the fall one.