"stick" anti personnel grenades fell out of favour due to the following.
The mills bomb and later "pineapple" type grenades were just cheaper and you could carry more.
Transport was also an issue.
However, anti-armor grenades still favour the stick handle, due to their requirement to be oriented so that the shaped charge is pointing towards the target.
This is accomplished in some cases by a drogue chute being stored in the shaft of the handle.
eg: http://en.wikipedia.org/wiki/RKG-3_anti-tank_grenade
See here for an example of it used in Iraq against armored humvees.
https://www.youtube.com/watch?v=r80bX0FOPnI
If I could follow this topic with a question: is there any truth to the idea that American style grenades were round because our troops could throw it like a baseball?
Stick grenades date from the first world war. Lots of good information about grenades in this thread.
The handle on stick grenades makes them much bulkier than other grenades. The only advantage the handle offers is that it makes it easier to throw it farther.
There were also a lot of improvements in grenade fuse design that occurred during WW2. Most stick grenades used a friction-type fuse, which worked much like striking a match by pulling a string. There were some that had percussion fuses, but even those required removing the end cap from the stick and pulling on a string. There were other types with designs so uncommon that they aren't relevant to this discussion.
After many years of experimentation, by the end of WW2 the type of percussion fuse with the pull-pin and lever that we are familiar with had become obvious as the most practical and easy-to-use design. The fact that you could carry two or three of these in the same space as one stick grenade made the choice obvious.
Stick grenades continued to be used, however. China made a grenade modeled after the WW2 Japanese type 98 stick grenade through the 1960s, and they were also used by the North Vietnamese. Also, many of the Vietnamese-made grenades encountered by US soldiers in the Vietnam War were stick types with friction fuses.
Nowadays, they aren't usually used as antipersonnel grenades, but stick types do turn up as anti-armor grenades, such as the Russian RKG-3 and its eastern bloc and Chinese copies.
Edit: I want to add that anti-armor grenades like the RKG-3 are not simple friction-fused grenades like the types common in WW2. These have shaped-charge warheads and complex and reliable impact fuses built into the handles. I brought them up here only because they are current examples of "stick" grenades.
As a follow up question, have sticky grenades ever seen widespread real-world use? (Ex. the video game Halo's Plasma Grenades)
The problem with stick grenades was/is the size. You could easily take six baseball grenades to a battle or you could take two stick grenades. The only real advantage of the stick grenade is the throwing distance. Soldier preferred more firepower than more distance.