The best tool for delicate work is a straight razor blade. Make sure you understand that the box cutter is the most dangerous tool on any job site before tackling the job yourself.
A box cutter like this will make short work of fresh thin-set.
Photo: Box Cutter - Image Source - Page
Getting the sides of tile clean is important before a grouting job. Ideally the sides are all clean and free of thin-set. Ideally you have at least the depth of room for grout as you do the width of the grout joint.
For a 3/8" tile set with a 1/8" grout joint I would like to see personally a good 1/4" of depth.