Is there anything more in his being "crucified", than if he had died some other death?

Yes there is; for thereby I am assured, that he took on him the curse which lay upon me; (a) for the death of the cross was accursed of God. (b)

(a) Gal.3:13 Christ hath redeemed us from the curse of the law, being made a curse for us: for it is written, Cursed is every one that hangeth on a tree: (b) Deut.21:23 His body shall not remain all night upon the tree, but thou shalt in any wise bury him that day; (for he that is hanged is accursed of God;) that thy land be not defiled, which the LORD thy God giveth thee for an inheritance.