The comments actually clearly put forward the complexity of a serious answer. But I'ld like to go simple and put mine for your example filters.
As you've stated, if two IIR filters of $70$th order and $4$th order (of whatever type) can satisfactorily do the job for you, then you should select the $4$th order filter because it will take much less processing power and processing time to compute its output.
That being said, those two filters (the elliptic and the Butterworth) have different frequency domain characteristics. They both have their own magnitude and nonlinear phase responses.
The Butterworth filter will have a monotonic frequency response in both of its pass and stop bands therefore in the design process it will be an overkill to specify a maximum allowed error ripple (from the ideal response) which will be satisfied at the cutoff freqencies but otherwise will be oversatisfied on the rest of the band due to monotonicity of the magnitude curve. That's why the Butterworth filter requires a much larger order for the same given filter specs.
The elliptic filter is known to have equiripple oscillatory characteristics on both the pass and stop bands which distributes the error energy accross all of the bands, therefore reducing the maximum (peak) of the allowed error ripple for a given filter order or minimizes the order of the filter for a given allowed peak ripple and transition width.
Since you don't want to consider the effects such as coefficient quantization or implementation architectures for high order recursive filters, then there isn't much remains to be considered for those two specific filters.