No, you can’t — some of those cards have been changed (see also the comment below), and you can’t experience them as they used to me.
What’s more, with the insane ‘power creep’ that’s apparently present in Wild (I don’t play it, hence this little disclaimer, but that’s what I see), those original cards would probably be laughable in terms of competitiveness.
PS Yeah, I appreciated it that Chillwind Yeti was considered one of the best minions stat-wise in the game, for example.
One more thing: alright, some of the cards in Classic were not exactly the same, let me self-quote a little:
Besides, there was this bug:
While reading through that topic, I also rememvered The Beast, by the way.
But still, despite these minor cosmetic differences, the game still mostly was and felt the same. Even those original battlefields — for me it was nice playing on them again.
PS Of course, there’s also Jaina’s art:
https://hearthstone.fandom.com/wiki/Jaina_Proudmoore#Patch_changes
https://hearthstone.fandom.com/wiki/Patch_13.2.0.28855#Undocumented_changes