If the result you are looking for is a photograph, the short answer is no. The same equipment won't give better results with video than it can with stills.

I think the apparent difference is due to exactly one thing: resolution. Try taking one of your still frames, resampling down so it's 1080 pixels tall, and then comparing.

There isn't any inherent low-light quality advantage in the way video is captured — in fact, it's really much more limited. But lower resolution hides many faults. Noise will be less apparent, but it also can hide motion blur — for example, from the longer shutter speed that you used for video.

Correspondingly, when inspected very closely, the still frames may seem bad, but if you actually make prints — especially relatively small 4×6" prints, but probably even larger ones — they'll look surprisingly good.

As @jrista notes, if you're not looking at single frames of the video but are watching it, motion plus our brains equals even more concealment of flaws. But, not so good for framing and hanging on your wall. Don't underestimate the power of that. While Youtube and the like make it easier to present short videos, maybe for sharing with grandparents and far-away friends, unless you put the effort into producing an actual film, the video file is likely to be forgotten, possibly dusted off for a graduation party or wedding reception video, but mostly not remembered. A photograph, though, can be an heirloom.