What is something the Spanish royal family does different from the British royal family?

They follow different religions which brings with different customs. The British are Anglican/Church of England, the Spanish are Roman Catholic.
Succession to the throne is now different. In Spain male-preference primogeniture means sons are ahead of daughters in the succession; Britain it is now absolute primogeniture by age rather than gender which is in line with most of the other European monarchies.
In the UK if the Queen does not approve the potential spouse of someone in the succession they lose their place. In Spain, both the parliament and the king would both have to disapprove a match for that to happen.

Spanish royals are Catholic; British royals are Church of England.

Almost all Spaniards are Catholics, not just the royals, so that's not something that really distinguishes them.

The Spanish do not have formal coronations on the scale of the British ceremony. The monarch is not actually crowned, though the crown is present at the monarch's oath-taking before the legislature (the Cortes Generales). This is in line with the practice of other European monarchies. There was no formal coronation with a crown placed on the monarch's head when King Willem-Alexander of the Netherlands took the throne.

Both countries are parliamentary democracies with a monarch as 'head of state' and a Prime Minister as political head. One difference is probably the relationship between the crown and the church in the two countries. In the UK, the Queen is Governor of the Church of England, the senior cleric is the Archbishop of Canterbury. Spain is a Roman Catholic country and the crown does not assume the same authority in the church.

I'm sure they have many differing traditions - and they're very different people. All families do things in different ways.

Given this involves people from two different countries, I don't think any differences would be confined to Royalty. The fact that the UK is a C of E country and Spain, Roman Catholic may make a small difference but again that would involve all people of those countries not just the RFs.

Etiquette/good manners covers people of all countries however.

British sit in fancy gardens sippin tea

Idk about Spanish

There is more tortilla soup and singing and that is all I am going to say.

The Spanish Monarch and family go to Catholic church and genuflect. We don't do that over here.