We consider the problem of computing a ((1+\epsilon))-approximation of the Hamming distance between a pattern of length (n) and successive substrings of a stream. We first look at the one-way randomised communication complexity of this problem, giving Alice the first half of the stream and Bob the second half. We show the following: (1) If Alice and Bob both share the pattern then there is an (O(\epsilon^{-4} log^2 n)) bit randomised one-way communication protocol. (2) If only Alice has the pattern then there is an (O(\epsilon^{-2}\sqrt{n}log n)) bit randomised one-way communication protocol. We then go on to develop small space streaming algorithms for ((1+\epsilon))-approximate Hamming distance which give worst case running time guarantees per arriving symbol. (1) For binary input alphabets there is an (O(\epsilon^{-3} \sqrt{n} log^{2} n)) space and (O(\epsilon^{-2} log{n})) time streaming ((1+\epsilon))-approximate Hamming distance algorithm. (2) For general input alphabets there is an (O(\epsilon^{-5} \sqrt{n} log^{4} n)) space and (O(\epsilon^{-4} log^3 {n})) time streaming ((1+\epsilon))-approximate Hamming distance algorithm.