19 is intended for height 5-9 to 6-3, 20 is intended for height 6-0 to 6-6. The 20 has a 1" longer seat tube (obviously), a 0.5" longer top tube (24.0 effective vs 23.5) and a 2cm longer head tube (15 vs 13). So the standover on a 20 is also almost an inch higher too.
So 19 is really the size for you if you're 6-1, but say somebody of 6-1 had relatively short legs and long torso, then (paradoxically perhaps) maybe a 20 would be better for the longer top tube.
But really it depends whether you like a big bike or a little bike under you. Even the size 18 is intended for up to 6-0, so if you like the feel of a small frame, don't rule that out either.