Aside from the fact that they're two different types of grain, how do the two malted grains differ in terms of their effects in beer such as flavor, body, head, etc. when used as base malts?
Assuming we're dealing with just basic malted wheat, and plain ol' 2-row malted barley...
Your malted barley has a clean smooth lightly malted flavor. It has enough diastic power to convert itself and other adjuncts, up to 10% of it's own weight. It is relatively low in protein, and easy to mash with a single infusion. Barley can be used for 100% of a mash.
Malted wheat is going to have a slightly sweeter taste. It should only be used for up to 50% of a grain bill, and typically needs to be mashed in the presence of other grains to help in lautering and prevent stuck mashes.. Wheat malt can make for a sticky mash that is hard to lauter, and makes for a very slow run off. If I brew with more than 25% of malt for my grain bill I always throw in rice hulls to help.
In homebrewing using wheat as a base malt is a pretty big pain.
Just complementing the answer above: Wheat malt also increases the body. It greatly improves the head retention, even in small quantities (5-10%). May result in a slight white haze due to proteins in suspension.
When brewing with wheat, it is common to add a protein rest (between 45 and 55 degress Celsius) for breaking down proteins into smaller proteins and amino acids. This will make the mash less sticky, and the resulting beer will be more clear. An extended protein rest will negatively affect the head though.
As stated in other answers, wheat alone is not a common base malt. Different sources recommend a maximum from 50 to 80% wheat in the grist. It is theoretically possible to use 100% percent wheat though, just be sure to add rice hulls for lautering.