In many countries including the UK and the Commonwealth, including New Zealand and Australia, the laws use the term ‘trade mark’, two words.
In the United States it’s ‘trademark’, one word, in the legislation. The American one-word form is increasingly used world-wide, which is why we tend to see ‘trademark’ in the press.
The International Trademark Association uses the ‘trademark’, American-style.
The Madrid Protocol talks about ‘registration of marks’, covering both bases.
Trade Mark or Trademark, it’s the same thing when it comes to protecting your brand names and product names.