I think you are talking about the minimum euclidean distance receiver, the two receivers you mentioned the Correlation receiver and the matched filter are 2 implementations of the minimum euclidean distance receiver. sometimes they are also known as ML receiver. The output of the two implementations will be the same at a particular sample say t=(n+1)Ts. If only the value at the sampling instant is of
interest, then the matched filter and correlator. give the same result, as has been already noted. But, intermediate results are quite different .
Matched Filter and correlation filter are both optimal filters . Normally matched filter is used for single wave form such as radar while correlation filters are designed to optimize SNR at the output. Both are called optimum filters in terms of SNR at the output.
Both the matched filter and the correlator maximizes the pulse SNR at the sampling instant in their outputs, thus they are optimal receivers in this sense. They are different devices (one is indeed a filter and the other is a multiplier followed by an integrator) and the waveforms in their outputs are usually different from one another, but at the best sampling instant, and only in this instant, the measured SNRs are the same in both