People object because although wearing the bearskin is a tradition, it is also a cruel way of obtaining material for a hat when there are synthetic materials available.See the following:-
A bearskin is a tall fur hat worn as part of the ceremonial uniform of several regiments in the British Army (most notably the five regiments of Foot Guards), the Canadian Army (Royal 22e Régiment (The Van Doos), Governor-General's Foot Guards, Canadian Grenadier Guards, The Royal Regiment of Canada) and by the Royal Life Guards (Den Kongelige Livgarde) of the Royal Danish Army. Until 1914 bearskins were worn in parade uniform by the Regiment des Grenadiers/Regiment of Grenadiers of the Belgian Army. The modern regiment has recently readopted this headdress for limited ceremonial purposes, although it is now made of synthetic fur,
The standard bearskin of the British Foot Guards is 18 inches tall, weighs one and a half pounds and is made from the fur of the Canadian black bear. The British Army purchase the hats, which are known as caps, from a British hat maker, which sources its pelts from an international auction. The hatmakers purchase roughly 100 pelts each year at a cost of around £650 each. Proper maintenance of the hats allows them to last for decades. Some bearskin hats in use are more than 100 years old. In recent times, attempts have been made to produce a version with synthetic fur, with little success due to the effects of the weather and static electricity. People for the Ethical Treatment of Animals has protested the continued use of real fur.
Traditionally, the bearskin was the headgear of grenadiers, and is still worn by regiments of grenadiers in various armies. However, following the Battle of Waterloo and the action in which they gained their name, the Grenadier Guards were permitted to wear the bearskin. This tradition was later extended to the other two regiments of Guards (Coldstream and Scots). The officers of Fusilier regiments also wore the bearskin as part of their ceremonial uniform. The bearskin should not be mistaken for the busby, which is a much smaller fur cap worn by the Royal Horse Artillery and hussar regiments in full dress. Neither should it be confused with the similar but lower racoon skin cap worn by other ranks of the Royal Fusiliers.
However tradition is one thing but to kill an animal un-necessarily just to wear it's skin seems sadly out of step in today's modern society. The arguement that bears are culled anyway is irrelevant. If the skins of culled bears are used do we also eat their carcasses?
Don't forget though that the British Royal Family is very pro-hunting and thinks that killing animals for fun is a jolly good pastime.