all grain is just that, all grain. partial mash is some grain and some LME/DME. Most of the fermentable sugars in a partial mash still come from the LME/DME while all the sugars in all grain come from the grain.
All grain uses more grain in a batch and generally requires a larger kettle to work with. I'm sure some of the AG brewers here can chip in some more differences.
Both are about the same amount of time/work, but AG requires some equipment beyond what extract or PM brewing requires: most notably, a kettle large enough to boil 7 gallons or so, a wort chiller to chill it quickly, and a mash tun/alutering system of some sort.
The reason some homebrewers do partial mashes is that there are many grains that don't really contribute anything to an extract beer if just steeped--they need to be mashed with some amount of a base malt like 2-row malt in order for their sugars and flavor contributions to be extracted.
So PM brewing opens up the many more possible grains available to the AG brewer, without some of the expensive equipment required.
BTW, partial-mash is *not* the same thing as extract w/steeping grains. That seems to cause some confusion here sometimes.