Well, Lycra is a DuPont trademark for elastane, Spandex was also a DuPont trademark but because they failed to police and enforce their rights was invalidated, declared generic and allowed to be used by anyone.
In general usage however, isn't it simply that Americans use the name Spandex whilst we Brits say Lycra, a bit like they say Plexiglass and we say Perspex ?
(again both trademarks - like Hoover, Thermos, Cellophane, Alley key/wrench, Pilates, Yo-Yo, etc, etc)