When I have a customer from UK, who's a resident of UK -- does this change anything in terms of taxes? As opposed to a customer from any other country.
If you register then you have to charge vat.
It's not 100% clear, as if the company is not regarded as established in the UK then there is no threshold. The definition of established in the UK is not the︀ same as tax resident or incorporated in the UK.
If you register then you can deduct incoming VAT (that you pay to your suppliers), so of course you also need to charge VAT if you are registered. I couldn't find exactly where this is written in HMRC websites, but there are some more info here: https://www.qualitycompanyformations.co.uk/blog/voluntary-vat-registration/
You charge vat on all sales to all UK customers (also companies) after you've registered for vat, as long as your product is subject to vat, and the place of supply is the UK.
You should check on place of supply of services for further details.