In UK law, gender reassignment is when you start / complete the process of 'transitioning' (if you believe such a thing is possible!). Someone who has or is going through that process has the legally protected characteristic of 'gender reassignment' and discriminating against someone for this reason is illegal in the UK. It doesn't require you to physically, medically or legally transition, but it does require you to live full time 'as' a member of the opposite sex. It only refers to men who think they're women and vice versa.
Gender identity is even vaguer and more nonsensical than gender reassignment.
Someone's 'gender identity' can be fluid, temporary, and importantly covers a myriad of supposed 'genders' that are neither woman nor man. Including but not limited to non-binary, gender fluid, gender queer, two spirit, non-binary trans (!), trans...... there were something like 78 at the last count, but as people are making up their own the list seems to grow by the day.
Many people question the concept, with good reason imo. It assumes everyone has a gender identity and that I, for example, 'identify' as a woman. This is simply untrue. I don't 'identify' as a women, I just AM a woman, always have been, always will be, and nothing I ever do would or could change that. But the implication / insistence that everyone has a gender identity is part of the move to normalise the views of gender ideology and draw everyone into the gender woo game.
Insisting on the idea that 'everyone has a gender identity' is like the insistence on announcing your pronouns, which is, btw, illegally compelled speech in contravention of your human rights. But making everyone put pronouns next to their name draws everyone into the fantasy life of gender ideologues (most of whom even aren't trans.)
It's a very good question. There's a lot of confusion about the terms - not least because Stonewall has been (mis)training organisations and telling them that gender identity is a protected characteristic under the Equality Act 2010. It isn't. The protected characteristic is gender reassignment, but this has a very specific meaning and the vast majority of people who claim some kind of identity other than man or woman do not fall into this category.