Nope, legally they are not the same, even though you're right, technically it was just a rename. But in the eyes of courts, Illusion ended without a legal successor, and Illgames owns none of their IPs whatsoever. (Meaning KK, KKS, HS, HS2, RG etc. are now free to use, there's no legal entity to sue you, Illusion is no more and Illgames has no copyright save HC).
The difference between renaming and ending+starting anew company is slim, but legally very very important (inheriting IPs and copyrights or not).