There is no physiological reason to not do both on the same day most military fitness programmes are built around doing both. With regard to the sequence there is continuing debate most of it anecdotal and not based on research. There is the do what is most important to you first claque which purely subjective.
Cardio and strength work use totally different fuel sources, strength training uses glycogen and cardio uses fats.If you do the cardio first you are burning up the premium fuel and you will not have the strength to strength training effectively.
The point which may be the critical one is that in the triad for weight control strength training is second in importance after nutrition and cardio is third. Strength training provides and increased metabolic acceleration after the workout is concluded,standard steady state cardio does not. The question now becomes , what are your goals?
Here is an interesting explanation for why strength should be first. www.buzzle.com/articles/cardio-training-fo
In my years in the military strength was always fist then cardio. In my years coaching track and field, cross country and cross country skiing my athletes always did their strength work first without any detriment to their sport specific training.
My clients do three strength training only workouts a week on alternate days and cardio twice a week on th non strength training days with a third at their option. All workouts are forty minutes or less including warm up and cool down, the strength workouts are circuits and the cardio is interval based. For weight control and general fitness if the quality and intensity of the workouts is adequate this is all the time you need to spend. Adding active recreation will round out you as an individual.
As an aside the Turbulence Training programme created by Craig Ballentyne ( www.turbulencetraining.com
largely forgoes cardio at all and his clients have sucess.