This is why glass enclosures are important in helping to stop heat loss. Rain can soak into the mortar joints, weaken them and, therefore, weaken the chimney. If you have a metal firebox, rain will cause rust. If you have a wood stove insert, rain will rust it rapidly. A cap will keep out birds and other varmints.
Bird droppings down the chimney can cause a bad smell and a breeding ground for mites.
Installing a chimney cap can prevent roof fires, as its spark arrestor will trap the hot embers. A cap inhibits downdrafting. Backpuffing of smoke can result from several factors. One of these is downdrafting, blowing smoke back down into the room. A cap keeps out leaves.