Step 1: Identify what makes ethane and ethene different.
Ethane, $\text{C}_2\text{H}_6$, has only a single bond between its carbons, while ethene, $\text{C}_2\text{H}_4$, has a reactive carbon to carbon double bond.
Step 2: Think of a reagent that reacts only with the double bond.
Bromine water is reddish brown because of dissolved $\text{Br}_2$. A double bond can open up and grab both bromine atoms in an addition reaction: \[ \text{CH}_2=\text{CH}_2 + \text{Br}_2 \to \text{CH}_2\text{Br-CH}_2\text{Br} \] and this uses up the colour, turning the solution colourless.
Step 3: Check what happens with ethane.
Ethane has no double bond to react with, so when it is bubbled through bromine water, nothing happens and the reddish brown colour stays exactly as it was.
Step 4: Turn this colour change into the identifying test.
Since one gas fades the colour and the other does not touch it, watching the colour of bromine water tells the two gases apart instantly. \[ \boxed{\text{Bromine water}} \]