How do I convert this data into a DiffEQ to predict values? I recently asked "Why do you need to solve a DiffEq?"  The topic was closed since too general, so I am opening with a specific question!
To recap, the answer given:  Say you had data for some variable as it changes in time, if you wanted to predict what the value would be in the future your best bet would be to estimate the derivative and solve the differential equations.
Personally, when I use existing data points to estimate unknown ones, I tend to use scatter plots, correlation, and a regression equation.  
I'm not seeing how this translates into DiffEq's.  Well, nothing illustrates/reinforces a concept better than a concrete example. Can we work through a simple contrived example? How about a situation modelling age vs. salary? Here are some data points: (age,salary in thousands) (20,30) (25,55) (30,225) (35,75) (40,100) (45, 110) (50,120) (55,130) (60,140) 
What do we do next to "estimate the derivative and solve the differential equations."  We can then predict income at age 65? or 43? 
 A: As hardmath explains, this example is not very well suited for motivating differential equations.
Here's how I see the lifecycle of a differential equation with perhaps a better example:
1) A physicist studies a natural phenomenon, and concludes that some quantity and its derivatives (rates of change) satisfy a particular relationship.
e.g.: A physicist is studying heat transfer, and observes (via experiment) that his cup of coffee is cooling at a rate proportional to the difference between the current temperature of the coffee and the ambient temperature of the room. She writes down this relationship as
$$
\frac{dT}{dt} = k(T - T_0)
$$
where $T$ is the coffee's temperature (as a function of time $t$), $T_0$ is the temp. of the room, and $k$ is come constant of proportionality.
2) The physicist would now like to predict what the temperature will be in an hour. She consults a mathematician who studies the equation abstractly, and is able to offer guarantees like "a solution to this equation exists" and "given the initial temperature of the coffee, the solution is unique".
3) The physicist and the mathematician now take the problem to an applied mathematician or programmer to actually get a solution. In this case, the equation is simple enough that the applied mathematician can state an exact solution:
$$
T(t) = T_0 + Ce^{kt}
$$
where $C$ is some constant depending on the initial temperature of the coffee. For more complicated differential equations, no exact solution can be written down, and the programmer has to use a computer to compute (approximate) solutions to the problem.
