Models, as you mention, are a huge source of applications. One can mention some in physics (free fall, radioactivity, pendulum, ...), in biology (cell growth), in chemistry (kinetics) among other. The beauty of the subject, tying together several branches of maths that student tend to consider separate, is of course also a strong motivation.
Let me stress three further points: a particular example in economics, the educational virtues of ODE, and its applications in mathematics.
The second law of capitalism
An example I like a lot (and used in my calculus lectures) is from Economics; the reason I like it is that it is a simple, static law which is better understood thanks to an ODE model. In Picketty's Capital in the XXIth century (where I read it) is stated as the second law of capitalism the relation
$$ \beta =\frac{s}{g}$$
where $\beta$ is the ratio of the stock of capital $C$ to the annual income $R$ produced in the considered economy, $g=R'/R$ is the economic growth and $s$ is the proportion of income which is saved, defined by $C'=sR$.
Now, this law is a limiting law, which may only hold true at equilibrium. But what if $g=0$? What if $g<0$? It then makes no sense whatsoever!
Now, one can consider where this law comes from. Assuming $s$ and $g$ are constant, since $\beta := C/R$, we get
$$\beta' = \frac{C'}{R}-\frac{CR'}{R^2} = s-\frac{gC}{R} = s-g\beta$$
Now, studying the ODE $\beta'=s-g\beta$, one sees that the only stationary solution is $\beta=s/g$, the second law of capitalism. But studying it in detail, one sees how the above absurd cases now make sense: if $g=0$, then there is no stationary solution, and $\beta$ increases linearly. If $g<0$, then there is no positive stationary solution and $\beta$ increases exponentially fast. Then, one can also study the speed at which the stationary solution is approached, which economically is important to decide whether the stationarity of $s$ an $g$ can be assumed (their characteristic variation time should be much larger than the characteristic time of $\beta$).
Functions as mathematical objects
The notion of function is a difficult one to grasp for student, and a central one. They have mainly learned to use them as processes, but at university we insist that they consider them as mathematical objects on their own. They have already applied processes to functions (e.g. differentiation), but we go much further (defining and using properties of function, operations on functions, relations on functions such as asymptotic comparison, etc.). ODEs are an important step in this process: we now ask them to consider functions as unknowns. Of course, other equations can do that (functional equations) but only few of them can be dealt with. (The very nice and short book by Alcock and Simpson was important for me to figure that out, I warmly recommend it). This is an important step on the path to further mathematical abstraction.
Applications in matheamtics
One should not forget that ODE are important not only on their own, but also as a crucial framework for much other mathematics. I am a differential geometer, so I need them:
to construct diffeomorphisms from vector fields,
to construct the exponential map of a Riemannian manifold,
to relate curvature and volume, ...
