A pilot departed ILM and because he missed a heading assignment as part of his takeoff clearance; deviated toward inboard traffic; which resulted in a traffic conflict.

Narrative

On departure from ILM on a Part 91 reposition flight; ATC cleared me for takeoff on Runway 35; landing traffic was on a 2.5 mile final. ATC says they issued my takeoff clearance on runway heading. After contacting departure and checking in; ILM Departure gave no new runway heading assignment and never repeated any assignment of runway heading or provided any traffic information. During the climb out; and after about 1500-2000 FT AGL; I started a slow turn to the right to join on course (direct which was how my clearance was given); this was about 20 degrees from my current heading. During the turn; ATC called and gave me a heading of 300 for traffic; they said traffic was on the approach going in to ILM. After a short vector on the 300 heading I was given direct. ATC gave me a phone number to call; which I did after arriving. The weather at the time was: KILM XX XA:53Z 33007KT 10SM CLR 28/21 A2985 RMK AO2 SLP109 T02830206. During the call; ILM Approach told me that the reason for the call was because of a conflict with other traffic landing in ILM. They said they reviewed the tapes and I was given runway heading as part of my takeoff clearance. ATC said that it sounded like I may have read it back; but that transmission was garbled so they were unsure. I told the Controller I did not remember any runway heading instruction. He asked for my pilot license number; address; phone number; etc. and said the info would be entered in the computer; but was not sure what would happen after it was submitted. I informed him that this would be filed on my part; and he said he would make a note in his system on that. The controller I spoke with seemed very cordial and did not seem upset over the event; just taking my information and moving on. The lack of separation from what I recall hearing was approx 2.5 miles.I believe that the event occurred due to rushing the takeoff; due to ATC informing me of landing traffic on a 2.5 mile final; and if the runway heading was given; I may have missed it due to hurrying the takeoff.