If a choice must be made, the food processor is more useful. Food processors are relatively lightweight and most of the functions are intergral to, or built into, the machine. Food mixers tend to be more cumbersome and their attachments take up much more space.
Food processors make light work of laborious tasks such as chopping, grating and slicing, and some machines also have juice extractors. They make excellent pastry, provided the ingredients are not processed for too long. They do not whisk egg whites well because they do not incorporate enough air, but some machines have a separate whisking attachment. Care needs to be exercised with some models when mincing raw meat so that it is minced, not pulped to mush.
Mixers are usually larger and are good for lengthy beating or kneading bread with the dough hook.