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Both series are heart warming slice-of-life animes involving students with alternate realities/world where their other "self"/"persona" fights. The characters struggle with the harsh realities of life and their inner feelings, developing strong bonds along the way.
Thay are also be tear-jerking, serious and lighthearted at different points, with gorgeous art. If you like one, you'll probably like the other.
As a side note, BRS is based on a Vocaloid PV and Persona 4 is based on a game; both adaptions are done well.
Although the games that both shows are based on aren't too similar, both Steins;Gate and Persona 4 are excellent examples of video game adaptations done right. Neither feel too bogged down in a video game style pacing, but don't jettison everything and start from scratch either.
Plotwise, both revolve around high school age students engaged in a mystery with science fiction/fantasy elements, and have a likable cast of characters. Both shows also manage to balance light-hearted elements with the darker mystery, and strike a very similar tone.