Relationship between order of filter and amplitude/phase response I am learning about digital filters, for example a lowpass butterworth filter. 

In many languages, the implmentation requires specification of the order of the filter, which I believe is the number of terms in the closed form equation that generates each filtered sample, given the original samples and previously generated samples. 

What I haven't been able to find (at a sufficiently ELI5 level) is how the order affects the frequency and phase response of the filter. 

Does a higher order simply mean a better approximation to the optimal filter with (cutoff, stopband) ? Do higher orders mean different or less severe phase differences?

Finally, I'd assume that it affects performance somewhat as well.
Does a higher order simply mean a better approximation to the optimal filter with (cutoff, stopband) ? 


That depends on how you define "optimal". In general a high order will give you a steeper rolloff


  Do higher orders mean different or less severe phase differences?


Higher order will result in more phase distortion


  Finally, I'd assume that it affects performance somewhat as well.


That depends on how you define "performance". The "best" filter always depends on the application. 

In general higher IIR filter order will give you more steepness in the frequency but also more time smear and phase distortion. Any beginner class on digital filter should cover this. For example, take a look at https://ccrma.stanford.edu/~jos/filters/