As far as I can discover, in 19th Century France, Chasseurs à Cheval and Hussars were both light cavalry with little or no armour, prioritizing speed over protection, armed with sabres and carbines, used especially for reconnaissance, raids and pursuit, although they could also take part in charges on the battlefield. They both had elaborately decorated uniforms.
'Chasseur' is the French for 'hunter', like the German Jäger, in a military context implying 'light' troops, who relied on speed to get themselves out of trouble rather than heavy armaments and armour.
Yet the two were considered to be different, and light cavalry regiments were designated either Hussars or Chasseurs à Cheval, not both. What was the difference and how important was it?