If the output value was only constrained by their sum being smaller than that of the inputs, you could e.g. set one output to a negative value and create new money in a second output.
You can also do more creative stuff: I think the rule might have been introduced as a response to the value overflow incident on 2010-08-15. A transaction in a (now invalid) block at height 74638 created outputs that were so large that their sum overflowed into the permitted range. The transaction created 2×92.2 billion bitcoins.