They are both part of the same group, Lloyds Banking Group.
They told me at Lloyds that they could not see my Halifax accounts. Halifax were part of my IVA. When I went in to close the Halifax account I had lying dormant for the life of the IVA I were told that they could see me Lloyds accounts. Go figure that one out!!!
I banked with Lloyds TSB throughout my IVA and did not have any problems, had the full current account only no overdraft, but had full offline debit card (the type that was contactless and could be used at pay at the pump machines) and cheque book.
I've since heard that if I was to apply for a mortgage with them they would check all accounts in LBG including Halifax. This has put me off them somewhat, so now moved main banking across to Nationwide who have no ties with passed creditors whatsoever.
To Lloyds TSB credit however, they did give me a credit card after the credit file got sorted out post IVA

We're willing to wipe the slate clean, give you a fresh start. All that we're asking in return is your cooperation.