Your internet router…

The prononciation ‘rowt’ is American

Yes. one of the things I had to get used to when I worked over there. Also:

Depot = deepow
Vase = vaze
Curtains = drapes
Tap = forcet
Cushion = pillow (but they also use cushion)
Single bed = twin bed
Duvet = comforter
Pavement = sidewalk
Take away = to go
Post or send = ship (regardless of mode of transport)
handbag = purse (but change purse for small purse/wallet)

Americans also often insert extra words where we would not find them necessary:-

(Me) I post next day
(American) I do ship next day

Two peoples divided by a common language

The prononciation ‘rowt’ is American

Router as in R-out.
Or router as in R-oot.

In the UK we pronounce router and rooter in the same way so not sure what this is about.

I don't understand the difference

This is a real bugbear of mine - you never hear Chuck Berry singing that he got his kicks on ROWT 66!

rooter here, routers go in the workshop

I understand that it's root-er in the UK, rout-er in the US. A rout-er in the UK is a woodworking tool !

Rooter

‘Rooter’ in this house, although spelt as Router