A HTML page encoded in Unicode is perfectly capable of displaying Chinese, Japanese and Korean (CJK) on the same page.
The problem is that most Chinese, Japanese and Korean pages are not encoded in Unicode.
Hong Kong and Taiwan tend to encode using BIG-5, Mainland China tends to use GB (means national standard), Japanese tends to use Shift-JIS (Japanese Industrial Standard) and I can't remember what Korea uses.
Note that each of these is a 16-bit encoding that copies 7-bit ASCII into the lowest 128 characters.
These different encoding are incompatible with each other similar to the way that French and Russian 8-bit extensions to ASCII were incompatible with each other (I had ex-Russian colleagues that often asked me to display emails for them that could be in any one of 5 different 8-bit encoding schemes, depending on who sent them, KOI was the most common).
A second problem is that Mainland China redefined the way they write characters in the 1950s.
The rest of the world continued using the traditional characters.
The traditional way uses many more strokes but in well defined patterns, the new "simplified" way uses fewer strokes but with less pattern to them.
Kind of like how some Latin based fonts have an "a" with just the plain circle with a straight line on the right and others add a fancy curve on top.
Except much more so and readers of one type have trouble reading the other without lots of practice.
Unicode doesn't solve this.
Partial solution is to have 2 fonts.
Both fonts display all characters but in the appropriate writing style - traditional or simplified.
Normally users in Mainland China have a font that encodes GB with the simplified glyphs.
Normally users in Hong Kong and Taiwan have a font that encodes BIG-5 with the traditional glyphs.
But there are some font files out there that do them in the opposite way so that a reader in Hong Kong can read a page from Mainland China (encoded in GB but displayed with Traditional glyphs) and vice-versa.
I spent a number of years in Hong Kong, China and Taiwan writing software for EFTPOS credit card terminals that had to automatically display English, Traditional Chinese or Simplified Chinese depending on information found on the users credit card (choosing the language was a black art but my code got it right for each of the card types that we had to cater for).