The differences between a partial mash and all grain are really pretty simple.
A partial mash uses a mash of some base grains (and usually some specialty grains) to produce some of the fermentable sugars, and uses some extract to add more fermentables.
All grain doesn't use any extract, so all the fermentables come from the mash.
Both methods can make excellent beers.
If you are going to do all grain, you need the capacity to:
1. Mash enough grains to achieve the required gravity at whatever efficiency you achieve.
2. Sparge the mashed grains to extract the sugars.
3. Do a full boil (although you could use some top-off water for lighter beers).
If you don't have the capacity for all grain, or if you choose to, you can mash a smaller amount of grain, add extract to bring up the gravity, and possibly do a partial boil, making up the final volume with top-off water.
So a partial mash gives you the ability to take advantage of all grain methods when you don't have the capacity for an all grain brew, or if you do have the capacity for all grain, allows you to reduce the time to produce a brew (because the sparge is quicker), or to adjust the gravity if your mash efficiency is lower than anticipated.
-a.