I cannot comment on the soaps. The aftershaves would fall into the same category of spicy scents, but IMHO (I just stuck my nose in a bottle of each) the "new" old spice aftershave is more perfumey than any bay rum I've used, that is to say that bay rum is spicier, but not as strong. They are quite different.
I have 3 Bay Rum aftershaves: Pinauld, Captains Choice, and Masters. 2 Bay Rum soaps: Col Conk and Honeybee
I have quite a few Vintage Old Spice products: colognes, aftershaves, talc, and shave soap.
I also have some P&G Old Spice products: Classic deodorant, bodywash, and aftershave.
I even have some Vijon Ivy Club Spice.
IMO, Bay Rum and Old Spice are quite different. But of course, YMMV.
They smell different. Old spice has a floral vanilla note. They compliment each other reply well though. I like to follow Superior 70 AS with OS cologne
I would say quite a bit different, this is based only on my products I make though. It may vary much depending on manufacturer.
I just recently made a soap scent similar to old spice, here is fragrance description: top notes are orange, sage, lemon that blends into carnation, geranium, cinnamon, jasmine and heliotrope, middle -bottom is vanilla, cedarwood, frankincense, tonka and musk
Now for Bay Rum: a blend of bay leaves, rum and spices.
Again, this is just from my personal knowledge, I am sure there are many different variations. I do think they can compliment each other well.