No, no they won't. Animal brain cells are too high maintenance and lack fine control. For neural net applications, we already have neural nets on-a-chip, and will continue to. They will simply become more complex, efficient, and scalable.

For non-neural net applications, they are far worse than conventional computers. Brain cells are terrible at math and precision memory.

Although this is just a spicy piece to attract grants, the research behind it is quite worthy. We won't be running programs on brain cells, but the basic research into doing so will likely advance brain-computer interfacing.