I think what may be causing your problems might be "screen tearing" on the LCD... LCD's run at a lower rate then CRT's (some do at least - most cheap LCD's definatly do)
If you enable FPS in game, then compare FPS read out on the CRT and the LCD, i bet they will be the same. Its just the carecteristics of your particular LCD might have Low response time.
If FPS read out is the same, try enabling 'Vsync' in your games advanced display settings on the LCD. see what happens.
EDIT: if you dont now how to enable FPS for example in CSS, just google for "show FPS in ________ <-game" its typically just a command such as (cl_showfps 1) or (com_drawfps) or whatnot, that needs to be input into console.