Interface is how your business looks like to other people. Interface is a sum of all interaction points where people meet you and your business – online, offline, phone, media, advocates, word of mouth, resellers, packaging and many other tiny details that form overall image of the business.
Classics of interface is the appearance of the shop, sales person, packaging, instructions, after-sales call if any, warranty service, complaint handling and so on. In a modern world online webpage, call-center, mobile application plays an important role. This is interface what businesses are controlling by themselves.
The other part is word of mouth, references, referrals, complaints in forums, blogs and offline community. There is a possibility to influence this information, but not to a full degree.
The difference between Interface and Channel can be easily explained by allegory of the house. If you as a business are inside the house (and the house itself as a part of business) – windows, doors, underground tunnel, chimney, phone line and internet connection is how you get yourself out. Interface is this house, its windows, doors, external entrance of the underground tunnel and the chimney from the outside – how people around see your business.