Whilst it would be an exaggeration to say there was animosity due to the historic rivalry of the two powers, relations between the men would fluctuate, and there was certainly a Napoleonic hangover amongst the command. The British commander Raglan who had fought under the Duke of Wellington against Napoleon was a belligerent man and somewhat senile throughout his later years (he would die in Crimea.) You can likely find many anecdotes on disputes, however perhaps the greatest breakdown in cooperation came in the spring of 1855 when the allied war ministers sought a change of policy.

Sevastopol was not fully encircled, and whilst its capture was central to the plans of all parties the French sought a change in policy as Napoleon III believed the town would not be captured until it was fully encircled. The allied ministers for war drew up three plans to invade the Crimean interior where it was believed their rifle power and support from the Tatar population would bring about swift victory. The memorandum allowed Raglan to select his preferred plan, but was insistent he embark upon one of them. In an act of insubordination Raglan called a council of war in the Crimea where he treated the memorandum as a mere "suggestion" and announced he would proceed as he saw fit. Raglan dragged his heels until April, whilst the French became increasingly impatient with Raglan's inaction, this sentiment can be observed in a message from French commander Canrobert (who had repeatedly offered to place some of his troops under Raglan's command) to Napoleon III - "The field plan worked out by your majesty has been rendered practically impossible by the non-cooperation of the Commander in Chief of the English Army."

The leaders would come into conflict once again when Raglan was compelled to move in March for a proposed assault on Kerch. The expedition set off on May 3rd after some delay from an initially reluctant Canrobert and sailed up the coast of the peninsula until a fast boat reached it with orders from Napoleon via Canrobert that the reserves were to be brought up from Constantinople and ships bound for Kerch would be recquired. The royal navy was forced to turn around and Canrobert was disgraced in the eyes of the British and French. The expeditions failure lead to in his view the undermining of his command, and feeling as though he had let Raglan down he resigned his post.

Source: Orlando Figes - Crimea