"The other kind" are called roper/roping reins, sometimes just loop or closed reins, depending. They are really designed for fast-paced western events (roping, barrel racing, etc.), and are not legal in the show ring unless you are riding in one of those events. If you're planning on showing WP, reining, cutting, etc., you'll need to get used to the split reins. If you're trail riding or generally pleasure riding, use the style of reins that is most comfortable to you. Other rein setup options are romal and mecate reins.
If your split reins can't drape over either side of the horse's neck and be out of the way without knotting them, they are probably the wrong length for your horse. Try longer reins if they are just too short to drape, shorter if they drape so far that the horse steps on them anyway.
Ultimately, it's a matter of personal preference. Some people are die hards for split or for loop, and there are valid safety reasons either way. Drop a rein with splits, and you may have a problem getting it back.
However, loop reins offer a hazard if the horse gets loose with them hanging. Choose the rein style that works best for you.