In the UK these terms could be used interchangeably, but often I would interpret the meanings like this:
Forward:
Person A sends something to person B, who receives it, but it needs to go to person C,
so person B forwards it to person C.
If I move into a house, and a letter comes addressed to a previous occupant,
and I know that person's new address (their 'forwarding address'),
I can forward that letter to them, by taking a pen, crossing out
the address on the front, writing the new address next to it, and
posting it (without paying for a new stamp). Also, for example, if I
accidentally damage someone's property, I might say 'Get it repaired
and forward the bill to me'. Perhaps I am involved in some legal matter, and I get a document which I can't deal with, so I forward it to my lawyer.
Redirect:
Person A sends something to person B, but person B has changed
address. The mail service can change the address on the item and
deliver it there.
If I leave an address, I can pay a fee to Royal Mail for
'redirection service' for a period, usually six months or a year.
Using a form, I tell them the old and new addresses, pay the fee, and staff at the area delivery
office serving the old address will (hopefully) grab any letters
arriving with my name at the old address, and change the delivery
address (using labels they printed) and put them back in the postal
system (redirect them). These never get to the old address.