Difference between applying voltage, and voltage across?
What Ignacio said is the core of the answer, I hope I can help you out going a bit deeper.

Generally the only distinction between "applied voltage" and "voltage across" is how you are dealing with voltage itself:


you apply a voltage to a bipole taking a voltage source and putting it in parallel with the dipole.
you usually measure a voltage across some dipole, putting a voltmeter in parallel with it.


That's to answer your question. Now what if you apply a voltage generator? What would the voltage across it? The answer is: there is no answer. That is a limitation of the model we are using. Ignazio makes the useful example of a diode: you apply 5V but across it there's only something like 0.7V: that's because your voltage source has an internal resistance where the remaining 4.3V drops.

Remember that most of the times when you apply a voltage to a dipole, the voltage across it will be exactly what you are applying. The two wordings though does not mean the same thing at all.

addendum

Since this is at the top now, and I've read some others very good answers, and since the question is very basic I'd like to add two words about potential, a word that every answer uses. A potential is a scalar field associated with a vector field. This vector field must be conservative for the potential to exist, and for the electric field this is true only for electrostatic fields. When things start moving around no potential can be defined. I don't want to be the fussy physicist but a professor once throw a chalk at me for this imprecision (he was quite precise) so since this might be seen from young students I though this should be pointed out.