Hi Dramaqueen,
You can use either or or both - depends on what you are serving it with! The dill works well with fish and a mint only, with pureed garlic works, well with a slow roast of lamb or BBQ lamb. Both work well when used as a dip If serving as a dip then it is a matter of taste. Personally, I use mint as a dip or both as a dip but dill if serving with fish.
For me, the critical ingredients are the yogurt and the cucumber. Find/buy the thickest Greek yogurt you can buy. How you handle the cucumber is critical. Top and tail and peel the cucumber. Divide into 4 and cut out the seeds. Grate, finely slice or finely dice the cucumber and place in a colander with a little salt and leave for about 30 minutes, rinse in cold water and leave to drain or dry handfulls in kitchen paper. It is important to remove both the salt and liquid. Failure to remove the salt might make the final mixture too salty and failure to remove the liquid will dilute the yogurt. Add the prepared cucumber to the yogurt together with the chooped herb(s) of you choice, stir well, taste and season with black pepper, EVOO and salt if necessary.
Greek yogurt may be used with just cucumber, prepared as above with or without garlic (cream using a knife or use a microplane etc), with mint, or dill or a mixture. The EVOO or OO should just be sufficient to cut through the acid of the yogurt but still leave an acid note or tingle on the tongue. If you go to far with the EVOO/OO add a touch of lemon juice.
I worked as a chef in Athens so my advice is informed by this - I have no doubt that others will disagree with me.
Hope this helps,
Archiduc