Hey zero_infinity and welcome to the forums.
The main advantage for analytic is that it's exact and gives you more context for what is going on. Having the equation can tell you something than just the output may not tell you.
For numeric the advantage is that you have to use this a lot since most problems don't have known analytic solutions, or that if they are known they are way too complex to deal with.
The numeric representation if its accurate enough tells us the same thing visually as the analytic model and for most purposes, this is what people need to see.
If you want to know where we currently don't have analytic solutions, search google for non-linear differential equations with no analytic solution or just get a book on non-linear partial differential equations.
In practice, it depends on the application. Some applications require really strict error control and this effects what models can be used and what the parameters are. Some are not so strict and just require that the output is good enough and stable.
There are also computational aspects to think about. It's not worth programming a computer to calculate a result that takes a week if you can do it in half a day with results that are still what you need. But sometimes if you can not trade-off accuracy, then you will need to use the best algorithms that do it the quickest even if that means waiting half a week.
These are some issues, but never the less important ones.