What exactly is the purpose of having an audio interface in live performance? Is there a notable difference between using the headphone jack and with audio interface?
When playing live, one issue is latency - the delay between pressing a key on your keyboard and hearing the sound start. To avoid this being a problem you need a low-latency audio driver, and many built-in laptop sound devices don't have official low-latency drivers. So that's one reason people buy an external interface, as any external interface marketed at musicians will come with low-latency drivers.
Having said that, your laptop may already have acceptable performance in this respect, or you may be able to get your latency down to an acceptable level with an unofficial low-latency driver.
Is the quality of the sound will be just fine? What type of cables are needed?
The basic audio quality out of a laptop can be good enough for live performance - but you need to try with your laptop; some have better audio performance than others.
It can be possible to get ground loop issues when plugging a laptop with an earthed adaptor into another earthed audio system. Solutions include running on battery while performing, or using a ground-loop isolator.
Laptops may have sub-optimal shielding and be liable to pick up environmental (or internal) EMF - again, this may or may not be a problem with your laptop.
if you are sure none of these are major issues, then a 3.5mm Stereo Jack to 2 x 6.3mm Mono Jack lead, such as this one (or a longer version thereof), may be appropriate.
One issue that may sound silly is the ease with which a 3.5mm plug can be yanked out of its socket accidentally; You might be at much less risk of this with an audio interface with sturdier 6.3mm output jacks (though of course you may then be at equal risk of a USB cable being pulled out!)