In what time of the year is there a high chance of having a visible aurora borealis in Sweden?
Are there significant differences between areas in Sweden regarding visibility of the aurora?
Obviously this will depend on cloud cover, so it is understandable that there is no certainty of seeing the aurora even if it is happening, so I would plan to be there for several days.
I researched this a little bit last year when planning a trip to Norway during which I saw several auroras. In theory, the best is around the equinox but you should really go in the middle of the winter, mostly to have as much darkness as possible. There is a multi-year cycle but the peak was a few years ago and waiting until 2025 or so is not very practical advice. There are also short-term fluctuations that cannot be predicted reliably enough to be useful for planning (but get an app to get an alert while you're there).
As far as location is concerned, there is a kind of ring or doughnut around the Arctic where auroras are most visible but anywhere in Northern Scandinavia should be good. The biggie is really cloud cover. Planning several days to maximise your chances is good but also think about local climate differences: Local topography (mountains, distance to the sea) can make a significant difference. Compare potential locations using past weather data and decide based on this.
Geographically you want to be in the northernmost third of the country. The further north the better. Auroras in southern Sweden are few and far between. You also need to be away from any big cities to avoid light pollution. The three major cities Stockholm, Gothenburg and Malmö are both too far south and too light polluted for good auroras. Cities like Luleå and Kiruna have lots of nights with visible auroras every year, to give you an idea of where you want to be to see auroras. Make sure you have something to do if you don't see auroras.
As mentioned in the other answer, the theoretical best time for auroras is not exactly mid-winter but slightly before and after, but this variation doesn't matter much compared to the probability of clear skies, which is usually better the colder the weather is. January and februrary probably have the best probability to get that cold weather with clear skies.
Go do something touristy in an area with possibility of auroras, and if you see auroras it's a bonus. Otherwise it will be a slow, expensive and boring wait for nothing. The Jukkasjärvi ice hotel, the Jokkmokk market, or Skiing in riksgränsen are popular winter attractions in northernmost Sweden. Also consider places in northern Norway like Narvik or Tromsø which have very spectacular scenery. Note that depending on your preferred activity, you may want to wait a little longer into spring. The sun doesn't go up at all for a couple of weeeks in mid winter, making some activities tricky. For example, for skiing at Riksgränsen you want to wait until late march at least.
To summarize:
You'll have the best chance in Northern Scandinavia and by joining an aurora bus tour that will drive you to wherever the overcast is the least, whether that's in Norway, Sweden or Finland. Many such tours leave at 5 pm and return you to your hotel at 11 pm but sometimes as late as 3 am. The aurora is visible every day during fall, winter and spring above the polar circle provided it's not overcast. But when the solar activity is low, the aurora will be weak.
It's worthwhile to take good camera equipment with you. You should have a tripod and a remote control for your camera (or you can use a timer). If you arrive early at a good location, you can often already take pictures of the aurora when it's not yet visible to the naked eye. I've seen the aurora several times in Northern Norway, the very first time was on my camera screen of a picture of the sky just above the horizon in the East to North Eastern direction, it was a one minute exposure at ISO 800. The first time I saw it with the naked eye was an hour later when it appeared much higher in the sky.
