I found mini winglets just in B 737-200. What's the differences between mini winglets and the blended ones which Boeing uses in more recent versions? (picture source)
The one above starts at 0.4x chord from the leading edge. For a test program on KC-135 tanker, which I suppose was first application of the winglets, they used the type of winglets shown in the question:
Conceptually there is no difference in the "mini-winglets" and the "Blended winglets" While I think almost all the wingtip shapes are more or less of same efficacy, I can think of couple of reasons for variations seen among various aircrafts - One is patents taken for some shapes would force other OEMs to come up with different shapes. Second is weight of the wingtip vis-à-vis the reduction in fuel burn. Typically winglets reduce fuel burn by ~3-8% but in some cases, the increased weight can off-set majority of the fuel burn or even prove to be more than compensating for it, especially on shorter hauls. One may be able to make a lighter blended winglet using co-bonded, co-cured composite structure as compared to the short winglet mentioned in the picture which with say bolted joints might end up heavier. I think it would be difficult to say make any generalised comment on this.
Boeing 737-200s are comparatively old and the winglets developed for them are not by Boeing, but by a third party, as noted by b737.org.uk:
... 737-200..., fitted with mini-winglets. This is part of the Quiet Wing Corp flap modification kit which gained its FAA certification in 2005.
The other types of winglets in B737s- blended wing winglets in 737-800s, the AT winglets in 737 MAX and Split Scimitar in NG 737 (through there is some overlap here) were all developed by Boeing (with others) and are available from production units and also as retrofit.
Adding winglets to an existing design puts stresses on the existing spar ends that they weren't designed for. Fortunately, they're stronger than may be necessary at the ends due to mechanical implementation requirements, allowing some additional load. But a "full size" blended winglet should be designed in from the start to ensure sufficient strength, minimal weight and avoidance of aeroelasticity (flutter).
The mini winglets require fewer modifications on the wings, and the -200 being much older does not warrant large investments in fimproved fuel economy.
The pic above, from this Boeing article on the NG winglets, shows the work required for winglet retrofit, based on stress analysis at ultimate loads in various aircraft states. The winglet shape basically extends the wingspan (but the weight helps to reduce the wing root bending moment), and creates aerodynamic lift and thrust as presented here.
The large winglet on the 737-800 is the result of a large and costly engineering effort, very particularly matched with the wing properties and the payload/range mission of the aeroplane. To make such a design effort for the -200 with its much higher fuel burn rate would be cost prohibitive.
The small winglets on the -200 pick the low hanging fruit of fuel burn reduction by reduction of wing tip vorticity. Same as the original winglets on the A320, from before they offered the Sharklet as an improved option. The deleted answer from @FeliceGraziono is correct:
Being smaller, they impart less efficiency but also stress the wing root less.