Small Aircraft; Low Wing; 1 Eng; Retractable Gear
Air Traffic Control Fully Certified
Situational Awareness; Time Pressure
ATC Issue All Types; Conflict Ground Conflict; Less Severe; Deviation / Discrepancy - Procedural Published Material / Policy; Deviation / Discrepancy - Procedural Clearance; Ground Incursion Runway
Air Traffic Control Issued New Clearance
Human Factors
Aircraft Y had landed Rwy14 and was rolling out to slow for parking. I then cleared Aircraft X for takeoff Rwy14. I realized almost immediately that I should have just put him in position and not cleared him for takeoff. I then told Aircraft X to cancel his takeoff and to Line Up and Wait. At this point Aircraft X still wasn't on the Runway yet. He didn't respond. I then asked Aircraft Y where he was parking; he told me; I told him to turn left next TXY and to contact Ground. Then I saw Aircraft X starting to roll on the RWY; I once again cancelled his takeoff clearance and told him to hold in position once again no response. I cancelled his takeoff 2 more times and never got a response. Aircraft Y in the meantime; had started to turn off the runway. Aircraft X departed and when I switched him to departure he responded. The pilot of Aircraft X had done the same thing on Ground Control; when told to stand by for his read back; he disregarded the Ground Control instructions and started to read his clearance back anyways. I don't know if he was having radio issues or if he just has a habit of not listening to proper instructions. I feel that I probably used anticipated separation a little too early; but I did catch it in time. I don't know if a procedure change is necessary; but maybe the pilot of Aircraft X; needs his radio fixed or needs more instruction on how to comply with ATC instructions.
Upon landing rollout on runway 14 at SRQ going through approximately 50 kts tower began repeating 'cancel takeoff clearance' to a departing VFR aircraft that was behind us on runway 14. The controller repeated the instructions 4 or 5 times with increasing tones of urgency; and we did not hear any responses from the aircraft the cancellation was intended for. As we cleared runway 14 to the left on taxiway J; I looked up and observed an aircraft climbing out from  runway 14; that was almost directly above us as we cleared the runway.  ATC never mentioned anything about the event to us and told us to contact  ground control for uneventful taxi and parking. Neither my FO or I remember the circumstances that the [other aircraft] was cleared for takeoff under; or if the [pilot] was cleared to depart at all. It is possible that the [pilot] was given a line up and wait instruction from SRQ tower but inadvertently departed out of habit; and was too focused on the takeoff (or radio volume too low) to respond to the command to cancel the takeoff from tower.  No evasive action needed to be taken; and we never heard anything concerning  a runway incursion from ATC.  I want to add that the [aircraft] was several hundred feet high by the time it got to taxiway J (about 7000 ft down runway 14) where we were clearing the runway.
Tower Controllers and a landing pilot reported another aircraft ignored instructions to cancel takeoff clearance and departed over the landing aircraft still on the runway.