The resistor is really cheap insurance.
The whole process of finding a current is based on assumptions. One of those is that the numbers you provide in the calculator inputs describe an actual point along the operating curve of the LED, and that is your target LED current.
For example, entering 3.3V and 20 mA implies that you know that the forward voltage is 3.3V
at 20 mA drive current. Not just that those are two numbers you found on the spec sheet. (Maybe you already understand this, so sorry for a bit of review if that's the case.)
Now the kicker is that this single point on the operating curve of the LED is
only valid at some reference temperature, most often the junction temp at about 25C (room temperature). Your data may or may not include the specifics, but trust me, V vs. I in a diode is quite temperature dependent.
Now what happens when you light the LED up? It dissipates power. How? As
heat. Immediately, the junction temperature changes. And what happens next is that the V vs I curve shifts.
Okay, you've stayed with me this far, let's look at what happens next with and without a series resistor (did I mention resistors cost a couple cents?).
Without a resistor, the current in the LED increases as temp increases. In turn, power dissipated increases, warming the LED up even more. This leads to a condition called
thermal runaway. Even if your voltage source is precision regulated to 0.01%, it doesn't matter. There's no mechanism to limit the current.
With a resistor (the smart choice), as the current increases, the voltage drop across the resistor increases proportionately (V = I*R, R is the constant of proportionality). This effectively (and cheaply) damps the thermal runaway, as long as that resistor is chosen appropriately.
This leaves us with one more possibility, which is the one of seriously underdriving the LED. The only time I think it's okay to omit the resistor is if your drive current target is far beneath typical, and thermal management is adequate to solve the risk of thermal runaway. In this case, the power dissipated in the LED is conducted away and not causing a threat to the junction. Most people like to drive their LEDs pretty hard though (brighter is better, right?). So just use the resistor. It's two cents well spent.