Difference between multinomial logistic regression and artificial neural network (multilayer perceptron) without hidden layer

Initially, they seems to be the same: they both have sigmoids within and NxK parameters where N is number of inputs, K is number of outputs. However, regression uses so-called softmax function, so all outputs sum to 1. The difference comes when one needs to run optimization routine. Logistic regression optimizes cross-entropy and ANN optimizes mean squared error. One takes a partial derivatives of these equations which result in slightly different formulas for regression and back-propagation. See more at http://ufldl.stanford.edu/wiki/index.php/Backpropagation_Algorithm and http://ufldl.stanford.edu/wiki/index.php/Softmax_Regression.