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The classical being out of tune is probably just the nylon strings stretching. Nylon strings can stretch for days.
As far as the physical differences, the classical guitar will usually have a perfectly flat fingerboard and the steel string guitar will usually have a radius to the fingerboard. The nut on the classical should be wider. I don’t see why you wouldn’t play both depending on your mood. The classical would be a good platform for finger style, and you have your steel string for anything you want to do with pick in hand. Not that both guitars won’t do both.
My Dad plays his classical almost exclusively, but absolutely never uses a pick for anything. He plays blues, strumming songs, the whole business on his classical.