The difference for me is that the meat of an algorithm is intentionally designed whereas the meat of a neural network is not. When training a neural network there can be some intention, but it's not the meat of the resulting "algorithm".

When thinking about this it's a bit like if I submit an application on fiver to write code for me that takes me from A to Z. I get the code back, I don't understand it and I didn't write it, is it still an algorithm? All of the same can be applied to a neural network.

Is natural selection an algorithm?

Is how the universe works an algorithm?

I think it's useful in every day life to distinguish what humans do and what something out of human control does. It can also be useful to be a bit philosophical and lump definitions together, but this only works if everyone agrees that they're doing this in a discussion.