a) The stochastic models are bottom-up or mechanistic models which are built up by the modeller from first principles how something is known to be working. It will include e.g. nonlinearities to the extent that our physical understanding of the modelled system includes nonlinearities. Their simplest, typical case is a dynamical system, particularly an autonomous (time-invariant) system: harmonic oscillator, chemical reaction rate equation. When they are stochastic (and not deterministic), you encounter them as Markov processes (time can be both discrete and continuous).
A stochastic process is by definition a collection of random variables, indexed by time typically (sometimes by space). Whereas in elementary statistics, you have independent, identically distributed random variables, the point of a stochastic process is that the variables are dependent (with some property stipulated about this dependence, e.g. Markov property or martingale property or stationarity).
`Martingales, Markov dependence and stationarity are the only three dependence concepts so far isolated which are sufficiently general and sufficiently amenable to investigation yet with a great number of deep properties.'
Loève, Ann. Probab. 1(1), 1973, p. 6.
This was later quoted in the Preface to Karatzas, Shreve, Brownian motion and stochastic calculus (2nd ed.), Springer, New York, 1998, so it was considered sufficiently up to date at the time.
I think we could extend it with processes indexed by space: random fields, Ising model, percolation, random graphs etc.; or by both time and space: interacting particle systems, asymmetric simple exclusion process (ASEP) and so on. Look up the table on Wikipedia about stochastic processes. These are examples of stochastic models.
Fitting a stochastic model to data is one kind of a statistical inference problem (model selection and parameter inference). So a) and b) are not disconnected areas.
b) Statistical models tend to be top-down or phenomenological. We don't really know what the underlying process (of the previous, stochastic process type) is that generated it. The best we can hope is to roughly describe how response variables react to changes in independent variables by any formula that seems to fit. Of course it helps to understand the underlying process to some extent. E.g. get the directions right (is something monotone increasing or decreasing?); recognise clear nonlinearities between the independent variable and the observation and model them with a nonlinear model.
Examples include (linear, generalised linear) regression models, autoregressive models (which model time series data but more superficially than a stochastic model).
It is also worth having a look here: What is the difference between a mechanistic and a statistical predictive model?
