What is the difference between an acoustic cutaway guitar and just an acoustic guitar?

Nothing really. A cutaway just means a difference in body shape that lets you have easier access to the higher frets.

When you hear the term "cutaway" in reference to a guitar - electric - acoustic or otherwise - it refers to a modification of a basic quitar design at the top of the neck nearest the body. The standard guitar shape is the "dreadnaught" design. People will interchange terms like Spanish or such but the basic design is called a dreadnaught. This where the body is symetrical nearly like a Figure-8 shape. The std dreadnaught has a round hole under the strings in the middle. This can change to an "F" shaped hole above andor below the strings which iis typically called a western style but otherwise is th same. The are many other variations in body size - slight shape variations and depth of body as well as wood. THe body may also be solid as with many electric guitars or semi solid for weight reduction or acoustic advantage sometimes called semi-acoustic. You will see thin electrics with sound holes in them like Rickenbackers or such. The solid bodies such as a Fender Strat have no holes.

Knowing there are all sorts of assorted bodies one feature on those bodies is that some will have the body "cutaway" at theneck - normally - at least- on the lower pick-guard  side - where the nexk meets the body. This permits the players hand and finers to more easily reach the upper frets when playing higher notes.  This would be called a SINGLE cutaway. If there is a second cutaway also above the neck in addition to the lower cutaway - this is a double cutaway. I don't believe I have seen a single cutaway with the cutaway only on top - that would not be of any useful value.

so you can have no cutaway - single cutaway (bottom only) or double cutaway (top and bottom).  the cutaway can be designed into any guitar when it is being built - solid or acoustic. 

So when you refer to a cutaway you refer to this indent in the body at the neck. Acoustic refers to the hollow form with a sound hole of some style . A the 2 terms are not related otherwise.  

For a visual reference - google up Gibson Les Paul - note the single cutaway.  google up a Rickenbacker 330 or 365 and note the double cutaway and the arrow shaped sound hole - a double cutaway semiacoustic. If you look into the line of Guild or Taylor guitars you will see a good example of acoustic -no cutaways - and single custaway acoustics (round holes etc).

I hope this helps you to understand

Marty

The other two answers covered the basic point. A cutaway is part of the basic body design that is "cut away" to give easier access to the higher frets for playing styles that utilize a lot of playing on the frets that overlap the body. The cut-away DOES detract from the acoustic power of the instrument due to the necessary internal bracing modifications as well as the lessened top surface and internal air volume, so a cutaway is very much a trade-off between playability and sonority.

One point about acoustic guitars. The basic shape is NOT necessarily a dreadnaught design, which is very modern and made popular by Crosby, Stills and Nash (and later C, S, N and Young) in the early 1970s and quite distinct from the smaller rounded Spanish shape of such guitars as the earlier Martin "New Yorker" and Gibsons (such as the one used by Bob Dylan in his early days on the scene).

Another basic design that is still very popular is the "jumbo" design, which is far more rounded than the boxy dreadnaught style. Also, making a comeback are the smaller, but still full scale, "parlor" guitars, which were the standard size a century ago. Then there are variations of classical guitars, one of which is the Flemenco-style. Then, there are the archtop acoustic guitars, not so popular today but still used by many.

The "cutaway" style is cut away at the bottom of the guitar so you can go higher on the neck or fretboard of the guitar.

A cutaway acoustic guitar simply has a slightly different body shape which allows you to reach the higher frets more easily.  Guitarists playing lead parts / solos on acoustic guitar generally tend to prefer the cutaway, however dreadnaught and cutaway guitars do sound the same.  Don't forget to find out if you need a guitar with built in pickup if you are planning on playing live.

http://www.camdensounds.co.uk/6-string-acoustic.html will show you a good range of acoustic guitars.