Round tables are an important aesthetic option, with some clear advantages. They are more welcoming, versatile, social and less annoying for diners: the legs are less uncomfortable and because there are no corners, we can avoid painful encounters. However, they require a very special space and are not suitable for crowded parties.
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!