Context: Coming from a statistics background, I learned a bit of R, then a bit of Python for data analysis/science, then found Julia as the language I invested my time in. Over time I keep up with R and Python enough to know what's different since I learned them, but don't use them daily.
What I always tell people is the following:
If you are writing code using existing libraries then use whichever language has those languages. The NN stack(s) in Python are great, the statistical ML stack(s) in R are simple and include SOTA techniques.
If you are writing a package yourself, then I assume you know the core of the idea well enough to be able to write your code from the "top down" i.e. you're not experimenting with how to solve the problem at hand, you're implementing something concretely defined.
In this case, and tailored to your use, I would argue that Julia has more advantages than disadvantages, especially compared to R or Python. Here are a few comments:
1. Environments, dependencies, and distribution can all be handled by Pkg.jl, the built in package manager. There is no 3rd party tool involved, there is no disagreement in the community on which is better. This is my biggest pain point with Python.
2. Julia's type system both exists and is more powerful than that of Python (types or classes) and R (even Hadley's new S7(?) system). By powerful I mean generics/parametric types and overloading/dispatch built in. You can code without them, but certain problems are solved elegantly by them. Since working heavily with types in recent years, I find this to be my biggest pain point in R and I wouldn't want to write a package in R, although I like to use it as an end user.
3. New developments in scientific programming, programming ergonomics, hardware generic code (as in this post), and other cool features happen in Julia. New developments in statistics happen in R (and increasingly Julia), new developments funded by big companies happen in Python.
4. The Python and R interpreter start up faster than Julia. The biggest problem here is when you are redefining types, which is the only thing in Julia that can't currently be "hot reloaded" i.e. you need to restart Julia to redefine types.
5. Working with tabular data is (currently) far more ergonomic and effortless in R than Python and Julia.
6. Plotting is not a solved problem in Julia. Plots.jl is pretty easy and pretty powerful, Makie.jl is powerful but very manual. Time to first plot is longer than R or Python.
7. Julia has almost zero technical debt, R and Python have a lot. Backwards compatibility is guaranteed for Julia code written in >v1.0 and Pkg.jl handles package compatibility. If I send you code I wrote 4 years ago along with a Project.toml containing [compat] information then you could run the code with zero effort. (This is the theory, in practice Julia programmers are typically scientists first and coders second, ymmv.)
8. You can choose how low level you want your code to be. Prototyping can be done in Julia, rewriting to be faster can be done in Julia, production code can be done in Julia. Translating Python to C++ production might mean thinking about types for the first time in the dev process. In Julia, going to production just means making sure your code is type stable.