You get priority support for telephone banking and you get a dedicated account manager (not sure if it's a fully accredited banker). You are automatically classified as one of the highest tiered customers for them and can get preferred rates etc. on many other services. Aside from HNW individuals, they also traditionally cater to "expats", which is a fancy British word for high income white collar professionals/digital nomads who either travel frequently or have foreign residencies for business. Your Premier status is global; you can easily create accounts at overseas branches with the same Premier privileges and (possibly credit history transfer too) and fund transfers between accounts are free, though I believe their conversion rates are not necessarily as competitive as a proper brokerage like Interactive Brokers. HSBC's premier service has been around long before the likes of Transferwise.

HSBC operates a bit like a franchise, each country's branch has to abide by domestic regulations but internally customers with Premier status are basically VIPs and get expedited during account opening at overseas branches. If I remember correctly, they generally take into account all of your assets with the bank including e.g. loans and mortgages etc. when banking. They don't only look at your cash assets when conferring Premier status. So if you have a house or car loan with them you can easily attain Premier.