MTBF Mean Time Between failures and MTTF Mean Time To Fail are the same numerically. MTTF just considers the time to the first failure. MTBF is more to do with the average period between failures assuming the system is emmediately repaired and put back in operation so that any down time can be ignored.
They are derived from the exponential statistical model that assumes the failure rate for a device or system is constant. (Plenty of info on the web on this.)
This is supposed to be the case for a pure electronics system during its usefule life period. I.e the flat part of the bathtub curve.
if the system has mechanical devices like hard drives and fans that have wearout mechanisms then it becomes a different ball game. I do calcs on set top boxes as part of my job and just the fan and HDD can be half the total failure rate!
See my email on modern MTBF predictions using HTOL test data for semiconductors.
MTTF (Mean Time To Fail) is the reciprocal of the Hazard rate Function or Failure rate(?)of a component or module. 1/?t is the unit of time per failure of that component or module.