Wooden chairs, finished with a luxuriously quilted seat cushions, not only look pretty, they are fantastically comfortable as well, making them ideal for both formal and informal dining. The only aspect to be very careful about is the fabric covering the pads, as it will need to be washable. Nobody wants to deal with pesky pesto stains or tomato marks, long-term and family homes always seem to encounter spills.
If you have one formal and one casual dining space, you can really go to town with different chair designs for both. A casual spot will do best with sociable, comfy seating that naturally leads to conversation, whereas a formal setting is where you can play around with luxury fabrics. If you only have one dining room, you're going to need to combine all of these elements in one set of chairs, if possible!