The formatting can provide semantic meaning to the copied text that is lost in plain text.
Bold, underlined and italic text can convey many meanings in text (e.g. titles, foreign language words and emphasis) that is lost if the formatting is discarded. What's more, this can have a profound impact on content written, for instance, in Word and pasted into a CMS for use on the web. Authors can't always tell that their copy has changed style just by looking at it.
Colour also has a similar use; how many times have we received an email saying something like "my comments are in red", only to find the email's formatting was removed somewhere along the way?
While typeface rarely conveys any special meaning for the text itself, symbol/dingbat fonts often repurpose existing characters to represent symbols. I receive emails from Outlook users ending in the letter "J" all the time because the sender used a smiley face, and Office automatically substitutes it with a character in Windings.
Tabular information especially can lose its meaning when it is converted to plain text. The spacial relationship between cells is lost removing a dimension of the data.
In general, preserving the formatting of copied text is required to adhere to the principle of least surprise (in most cases). Where formatting is likely to be surprising (e.g. in Notepad or in InDesign), it should be (and often is) suppressed.