Why are RNN/LSTM preferred in time series analysis and not other NN?
I'll try to provide some insight which will hopefully help.


Can a normal NN model the time connections the same way like a RNN/LSTM does when it is just deep enough?


Every neural net gets better in theory if it gets deeper. For a regular NN to model time connections properly, you could use the last n time steps as your input and the n+1th time  step as your target. This will generate your training set and depending on your data, you could be able to model your time series fairly efficiently.

All of the most obvious pitfalls of this approach are actually addressed by RNNs/LSTM.


Does an RNN need more or less data in comparison to a NN to get the
same/ better results?


Difficult question, which would probably require some empirical results to check that theory. Also, with neural nets, sometimes, it's more about how fast it trains rather than how much training data it has that will make the biggest difference in performance.

To me, the main difference is that your regular NN will need a fixed-size input, whereas your RNN will be able to learn with input "up to" a certain size, which can be a big advantage to model the entire time series well.


Are there time series where normal NN or RNN/LSTM perform better?


Again, this is a difficult question as it will depend on the data, the architecture of the networks, the training time etc.

I haven't done empirical research on this, but I'd say that your best guess would be time series that are rather short (less than 100 time steps for instance). If the time series is short, you might not need to model such an intricate relationship through time, which a regular NN could perhaps do as well as an RNN.


Is it time data depended which Model will perform the best or are there some guidelines?


As explained above, every learning exercise will highly depend on the architecture and hyperparameter used, even the initialization of your weights, whether you use pre-training or not, and of course, your data. Again, I think that the shortest the time series, the more competitive regular NNs could be. But again, this is merely intuition-based and hasn't been checked thoroughly.

- Can the NN behavior be understood better than the RNN behavior?

It all depends on what you mean by understanding. RNN have more weights than NN so ultimately, there will be more things to analyze and eventually understand. But, with more data also comes more information, so perhaps there is more information to be gained from an RNN than a simple NN, even if it's a deep one.

Plus, sometimes, based on the initialization of the weights and other parameters, the interpretation of the model could vary, even if it's trained on the same data.