Almost all statistical tests start with an underlying distribution, and then find the most likely parameters. You first set up a "stochastic assumption" (we can use central limit theorem if sample size is large so lets use a normal distribution, the relationship is linear so lets use a linear regression). After making those assumptions, you go into the statistical process of getting maximum likelihood estimates, p-values etc. You check the values, reexamine the assumptions, and repeat until you are convinced.
The distinction comes from where you spend most of your time thinking about. And oftentimes where you spend most of your time comes from the use case of your model. Let me clarify through some examples.
Suppose there is a coffee shop. If you are a simple investor, you might get statistical estimates for revenue/profit, growth, expenses etc. and get your rough estimates for its future cash flows and put a price on it. Those simple statistics could be enough for you, because you are less interested in how to model the actual business, and more interested in getting some reliable estimates. On the other hand if you are a manager, you might think about constructing a more sophisticated stochastic model like an MMC queue, and then get estimated arrival times, service times, revenue per costumer, costumer loss per waiting time, etc. Look at marginal differences and take action accordingly.
In financial world, sometimes a small correlation could be a good buy/sell signal. For high frequency trading you might heavily use statistical approach and don't justify the underlying economics much, as long as it does generate revenue. A good example could be momentum trading where you trade on the signal, but it doesn't have a strong economical explanation aside from slow diffusion of news. Whereas for a deep prepayment model or equity research you might want to build a stochastic model, come up with random variables for every effect you think. Then use statistics to get the stochastic model values.
In short often times they go together, with stochastic models you "declare" your model of the world and use statistics to get estimates for your model. Depending on which part you put more focus on, we say stochastic model or statistical model to identify loosely.
Hope it helps