var words=textarea.split(" ");
This is Wrong!
Why? Because a space isn't the only thing that can separate words. If the user hits Return at the end of the word, this inserts a newline, which won't be counted as a word separator by the above code. And if the user adds commas or other punctuation that is not followed by a space - or adds a comma with a space either side of it - then this will also result in an incorrect count.
Even worse, if the user has more than one space between words, or extra space at the end of the input, then this will up the word count making it even more inaccurate.
Fortunately there's an easy way to count words using regular expressions. A pattern of '\w+' will match a string of characters that can be in a word. With this technique, there's no need to worry about trimming the string to remove excess space, either.
So try this instead:
alert(words+" words\n"+chars+" characters");
for a much more accurate result.