I'm surprised at how many people don't see a difference between "bite" and "bite into". If you bite something, you don't necessarily actually take a piece out of it. My friend had a horse bite her once - left a huge, nasty bruise, but didn't break the skin. It's the action of chomping down on something with your teeth.
To me, though, to bite into something is particularly food-oriented and means to take a piece out of something that is too big to eat whole - bite into an apple, a pork pie, a chicken leg.
Macmillan dictionary says they mean the same https://www.macmillandictionary.com/dictionary/british/bite_1 Except that 'bite into' has some other meanings https://www.macmillandictionary.com/dictionary/british/bite-into