The observed performances of carbon nanotube field effect transistors are examined using first-principles quantum transport calculations. We focus on the nature and role of the electrical contact of Au and Pd electrodes to open-ended semiconducting nanotubes, allowing the chemical contact at the surface to fully develop through large-scale relaxation of the contacting atomic configuration. We present the first direct numerical evidence of Pd contacts exhibiting perfect transparency for hole injection as opposed to that of Au contacts. Their respective Schottky barrier heights, on the other hand, turn out to be fairly similar for realistic contact models. These findings are in general agreement with experimental data reported to date, and show that a Schottky contact is not merely a passive ohmic contact but actively influences the device I-V behavior.