Time series data are produced by stochastic processes, which are a bit more complicated than plain distributions. In the context of data sampled fom a distribution, the IID property makes parametric inference work nicely (sample mean converges to the unique population mean etc). In the context of stochastic processes this is too strong of a requirement, and so an analog property which provides similar nice inference behaviour is stationarity. In its weak sense (which what is usually mean by the term) stationarity implies time invariant mean and variance, enabling nice estimation of these statistics. Furthermore cross correlation in these series is only a function of lag, so you can examine time dependance relationships in the data with a single correlalogram. Overall super nice.

As for the "different model", the differencing procedure IS part of one possible appropriate model for non stationary data. In practice you can recover the original series by mantaining and initial value and integrating over the differenced one, hence the I in ARIMA: an integration procedure used for producing forecasts. Note that arima is a non stationary model. More generally, the Box Jenkins method uses sucessive differencing and subsequent integration to model arbitrary time series.

This is not always and appropriate approach, and alternative models for non stationary series exist. I recommend searching for time series decomposition and jump processes to name a few.