The 38 special will be in a revolver and the 380 ACP most probably a semi-auto. However some of the energy of the bullet is used to eject and reload the 380 which dampens the recoil a bit. Also the .38 special is actually 0.357 " diam and the 380 is 0.355". The upshot of it all is that the .38 would have more recoil.
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There'll be a notable difference... part of the difference with a .380 vs. a .38 Special is that the .38 Special will be a revolver, whereas the .380 is much more likely to be an automatic, so part of the recoil felt with the .380 will be the movement and weight of the slide.