real is an adjective, it describes the noun. not a real bear = inte en riktig björn e.g. - it is just a toy. But really (actually) is an adverb, reinforcing something more general in the whole phrase, so in this sentence It is not really a bear = Det är egentligen inte en björn we comment on the whole situation. Maybe someone gets scared when a big dog comes running, and has to be reassured, or two people are watching TV and differ in the impression of the animal, is it a bear or a dog?, or a child let a dog 'play' a bear, and then declares in a grown-up way 'Actually, it isn't a bear, It's really a dog'. etc.
This reminds me of this video https://www.youtube.com/watch?v=ihpG_NJ_T1g&ab_channel=muxer107