The manga took a 3 week break because the mangaka took a break. A lot of the mangaka took a break during the 1st week in January to celebrate the New Year with their families, and took the other 2 weeks to rest up for the upcoming year. I am not 100% sure, but I believe the magazine Kuroko is published in, Shonen Jump, also took a 3 week hiatus to give its employees a break for those same reasons. There is a long break in the anime because after the 1st season of the anime ended, the 2nd season of the anime had not been approved yet. The announcement that the 2nd season got greenlit came out right before the new year, but since then there has been no further information given out about when the anime will start airing.