Because everyone else calls the thing a 'space elevator'.  Changing the basic name for marketing seems at this point futile and time wasting; might as well try and halt the tide.

"Spacelift," although proper as it may seem, would not be as hard hitting in the public eye as "Space Elevator."  Bdunbar had it right as well. In general though, both terms mean the same thing. An 'elevat'or  'elevates' as a lift 'elevates' as well. Then again, I am sure the public wouldn't delve that deep into the terminology.