A remortgage is (just like an original mortgage) simply taking out a loan with your house as the security for that loan. It can be useful for people who've still got some of their original mortgage to pay off but who want to seek a better rate, pay over a longer period or borrow more (possibly to finance some home improvements).
See here
https://www.moneysavingexpert.com/mortgages/why-remortgage
and here
https://www.moneyadviceservice.org.uk/en/articles/remortgaging-to-cut-costs
With equity release you take out a 'lifetime mortgage' where, just like with any other mortgage, you borrow money with your house as security. However you don't pay anything back. Your debt keeps rising as interest is added to it until, when you die, that debt is paid off through the sale of your house.
(There's also another form of equity release called 'home reversion', where you actually sell part of your home but remain entitled to live init. The downside of that arrangement is that the price you get for a part of your home is usually far less than its true market value).
See here:
https://www.moneyadviceservice.org.uk/en/articles/equity-release