Different rods have preferance for one or the other, material being welded, position of the welding to be done, there are many factors that arte involved. The difference is a stringer is a drag or push without any oscillation. A weave can have different shapes and patterns the trick is finding what works for you on the given job and material.

A text will aid in definition and your understanding,

A stringer bead is typically a weld with a width of 2-3 times the diameter of the electrode. (This applies to the SMAW process) Good for heat input control and placement. Used on most any joint type in any position.
A weave bead is usually 4-6 times the diameter of the electrode core. Used for high deposition rate welding where heat input is not a prime consideration. Not recommended for horizontal or overhead positions.

According to AWS A3.0-94 Standard Welding Terms and Definitions

"Stringer bead. A type of weld bead made without appreciable weaving motion. See Figure(22A)"... which basically shows a butt joint where the bead is made in a straight continuous motion with forward progress along the joint.

"Weave bead. A type of weld bead made with transverse oscillation. See Figure (22B)"... which shows a butt joint where the bead is made with a side to side motion as forward progress along the joint was made.

I agree with cccasey in his description of the two beads in that most people define them as he did in reference to the ratio of the bead width to the diameter of the electrode.

John Wright