Why the driving point functions have the condition that the degree of numerator and denominator only at most differ by one?
For very high frequencies the driving point admittance/impedance (real and positive) primarily can be only (a) inductive, (b) capacitive or (c) resistive. For this reason, the difference in degree between N(s) and D(s) can be at most "1". 

EDIT 

Remember: The degree of a function is determined by the highest exponent (s³ gives third degree). And - on the other hand - for very large frequencies it is the highest degree which dominates. For this reason, the answer is derived from circuit properties for frequencies approaching infinity. 

Because for very large frequencies (infinity) the function must be of first order only (capacitive or inductive), the degree of N(s) and D(s) must be  differ by "1" only (examples: s³/s² or s²/s³).