I have a data structure based on which I generate a dynamical behaviour that I want to integrate. So I construct a rhs.
I further want the user of the library be able to pass it new functions that can be integrated into overall dynamical behaviour.
There are different ways to achieve this, the simplest version is with closures. Pass a list of functions, and some parameters and I construct a right hand side function from it. Unfortunately this does not work with numba. What I ended up doing is passing not the function itself but the function text to generate the code of the function to be jited and then eval that. It worked but it was horrible to maintain, and required users to pass function bodies as text witha very specific format.
Now in Julia we will probably eventually transition to a macro based approach, but the simple closure based model just worked.
Previously I had large scale, inhomogeneous right hand side functions that I wanted to jit in numba and that need sparse matrices. So I ended up having to implement sparse matrix algorithms by hand because I can't call scipy.sparse.
Another instance: I implemented a solver for stochastic differential equations with algebraic constraints in numba, partly to be able to use it with numba jited functions and get a complete compiled solver out of it. This already constrained my users to use numba compatible code in their right hand side functions.
In order to get this to work I had to implement a non-linear solver from scratch in numba rather than being able to use scipys excellent set of solvers.
Julia is not a magic silver bullet. Getting the ODE solvers to make full use of sparsity still requires some care and attention. But I simply spend a lot less time on bullshit than before. (so I have more time to spend on HackerNews :P)
I decided to switch over when for one paper I was able to implement a problem using the standard tools and packages available in Julia within half a day. The Python equivalent would have involved using a new library that came with its own DSL, which would have meant rewriting quite a bit of my code to take advantage of it. Easily several days work.
With DifferentialEquations.jl I also could just test half a dozen different numerical algorithms on a problem in a matter of minutes, find out which performed best and use that for MonteCarlo. Saved about a week of computation time on one project alone. That's not a critical amount, nobody cares if the paper comes out a week later or earlier, but it's nice (and I don't waste super computer time). With Python libraries with different DSLs this would have taken considerably longer, and I probably would not have done it. This is the result of having one library and interface rather than a whole bunch, if everyone agreed on scipys ode interface (which just got properly established in scipy 1.0.0) this would be easy in Python as well. But that's also the point that people have been making: Julias design for composition over inheritance makes it convenient to rally around one base package.
I also personally very much like being able to enforce types when I want to. This is a big win for bigger projects for us.