As numerous others noted:
Sodium Tallowate is made from sodium hydroxide (lye), steam, and animal fat (tallow). So a tallow/ sodium hydroxide soap and a soap made with sodium tallowate is the same stuff. Sodium tallowate implies that
the tallow was saponified (see saponification) in a factory/ chemical factory and delivered in bulk to be used as an ingredient. It should also not have the risk of excess or insufficient amounts of lye that might exist in hand made soaps.
Sodium tallowate - Wikipedia, the free encyclopedia
I might add that the OLD soaps have a chance of fully reacting should it not have
when made. Even if the component ratio was identical, time can mellow such
reactions that take time even if accelerated by heat (steam).
As others noted this component is the same except for the possibility that the old formulation had a little excess lye that does soften hair. If someone has a pH kit for a pool and both old and news soaps it might add some info here. I thought I saw a pH kit for an aquarium that might work.