What makes DNA sequences most different/recognizable from a biological perspective?
This isn't a question with a really well accepted answer yet, and comes up quite a lot in e.g. studies of population variation in transcription factor motifs.

Usually, we approximate the sequence preferences of a DNA-binding protein with a position weight matrix. A weight matrix will given you a score for two sequences,  so the simplest means of quantifying the relative binding strengths for two sequences is compare these scores. You could also, say, compare the relative chances of getting the scores under some background distribution of scores, which would arguably be more comparable between different factors.

The PWM score (and other means of describing sequence specificity) are only an approximation to what you're really interested though, which is the binding energy associated with the interaction. If you have detailed experimental evidence about how the protein binds, you might use that, and there are some papers that build models for approximating it from PWM scores as well. However the reality is that you can only ever roughly approximate what's happening in the cell, because of molecular crowding, and the many other factors that will be present on the chromatin fibre, so the difference in PWM scores is often what gets used. 

Predicting the impact of a given change to a non coding DNA sequence is a hard problem. Sometimes you'll get a decent approximation, but in the end you're trying to do biochemistry from first principles, and we aren't there yet.