If you're able to install a hood which vents outside, then that's definitely the better option. Downdrafts do work to some extent but with limitations. If you have a gas cooktop, for example, the downdraft may disrupt the flames on the back burners. Also, because heat, steam, etc. rise, a downdraft won't catch everything, particularly from tall pots - although the telescoping downdrafts are better at this. You definitely don't need both a downdraft and a hood, though.

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Many posters here claim that a hood is the only way to go. And while a good hood and motor is the most effective it isn't the only answer.

By downdraft do you mean a popup vent or a JennAire?

I have a Dacor pop up vent with a remote motor and it works very well for us. Dacor offers 2 motors - 400 and 1000. Depending on the design of your vent run a 400 is fine unless you have a long run or lots of turns. Ours is very short with only one angled bend.

Our cooktop gets a lot of use and we never had lingering smells. Some posters say that the vent pulls the gas from the burner. We haven't had that problem. We do sear steaks on high heat in a cast iron skillet without any smoke or odor problems. Our burners are 18000 BTUs and we do stir fry often. Perhaps if you had a 23,000 BTU burners the hood might be the better choice.