The usual purpose of using an encoder at the front end of a network and a decoder at the tail is to deal with features in between. Between the networks is a representation of the input with much of the redundancy removed.
Some of the approaches are discussed in On the Properties of Neural Machine Translation: Encoder–Decoder Approaches, 2014, Kyunghyun Cho, Bart van Merrienboer, Dzmitry Bahdanau, ,Yoshua Bengio. A single network that, given inputs, outputs some value and later appends that value to the input to produce the next output is not an equivalent circuit. Without the separation, network cannot be trained on the basis of some characteristic of the unique features found at the middle tap point.
Those learning how artificial networks can perform simple tasks tend to draw the system boundary around a sequence of layers where the inner layers are not manipulated differently from a forward propagation and back propagation perspective. They will call that the network and the layers that are not directly manipulated as elements in the series of layers are called hidden layers, which is correct, but they are not really hidden, they are just not addressed at the higher level, at the system boundary. They're manipulated as elements in the series by the forward and backward propagation mechanisms.
Those who have linked these sets of layers together into larger systems sometimes use the term network in the wider sense, a circuit that contains more than one sequence of layers. That's typical in NLP and motion control systems literature. The difference is the complexity of control of the learning process. Some control schemes require the interrelationship of data paths that are not at one of two ends.
Many have attempted to reduce this complexity, but there are control theory reasons why some systems cannot be reduced without losing control of some aspect of the learning process that is directly tied to the objectives of the system. Take a look at textual natural language translation, speech synthesis, robot imitation, automatic pilots, and terrain learners to see examples where the assembly of multiple networks into the larger system are necessary.
Also, feeding outputs into inputs defeats the design of the recurrent or long-short network design, which handles the retention of state in a way that converges to minimize loss inherently.