DWH Controller described conflict event between IFR and VFR departures when the VFR aircraft maneuvered to avoid weather placing aircraft in direct conflict with the IFR aircraft; requiring evasive action to avoid collision.

Narrative

The weather was marginal VFR with low ceilings; but pretty good visibility. I cleared Aircraft X for takeoff; knowing that his flight direction would be West. I told him that a left turn to the West was approved; he never read it back; but I saw him start his turn and proceed westbound. I then got an IFR release on a Light Transport. I then cleared the Light Transport; 'turn left heading 290; maintain 2000; Runway 35L cleared for takeoff; proceeding aircraft is West bound.' By the time the Light Transport started his takeoff roll Aircraft X was two miles West of the field; so I turned to watch the Light Transport take off. When the Light Transport got airborne; I looked at the STARS to watch for a tag and I saw that Aircraft X had turned and was now due Northeast bound and climbing. I asked him right away what his intentions were and he said he needed to maneuver to remain clear of the clouds; but continued climbing. I called the Light Transport traffic to him; but he did not see him. The Light Transport pilot saw Aircraft X at the last second and made a drastic turn to avoid him. In a later conversation with the Light Transport pilot; he said that the only reason he saw Aircraft X was because he happened to be in between the cloud layers; and had he not turned when he did; he would have run right into him. The Aircraft X pilot; however; did not seem to notice. Recommendation; make sure I get a readback on everything. Next time in this same situation; I would probably have sent the Light Transport Southwest bound until he was clear; just to make sure this doesn't happen again. I will not hesitate next time to change the heading of the IFR to avoid collision.