I had a look at the difference between mnunuzi and mteja in a Swahili-only dictionary here https://sw.oxforddictionaries.com/ and I'll translate the definitions here.
For mnunuzi (usually translated as buyer) it says mtu anayepata kitu kwa kulipia kwa fedha "someone who gets something by paying for it with money".
For mteja (usually translated as customer) it says mtu anayekwenda kupata huduma au bidhaa mahali fulani k.v. dukani, hospitalini, n.k.. meaning "someone who goes to get service(s) or good(s) somewhere, e.g. in a shop, hospital, etc."
So, I guess mnunuzi is more or less a customer (someone who walks away with a physical thing) and a mteja can mean either customer or client (someone who pays for a service) ... at least as I generally use and understand these English words.
(k.v. = kama vile "such as", n.k. = na kadhalika "and such")
Thank you for this in depth reply. Based on my experience, mteja is used more commonly than mnunuzi, especially for someone walking around the market. Mnunuzi (wanunuzi) seems to be more commonly used when referring to wholesale buyers.