To elaborate on Mick's reply:
D-76 stock (full strength) has a high concentration of sodium sulfite, which acts as a silver solvent, dissolving and redepositing silver on the developing grains. This results in "finer," but less well-defined grain. At 1+1, D-76 has a lower concentration of sulfite, and the grain is therefore more pronounced, but "sharper."
With other developers that don't exhibit this kind of silver-solvent activity at full-strength, diluting often makes no real difference as long as the development time is proportionally extended and agitation is consistent and often.
However, many use dilute developers and longer periods between agitations to maximize compensation and edge effects. When the agitation is less frequent and the developer dilute enough, it exhausts in the areas of higher density (i.e., more active development) before the next agitation cycle. This results in those areas receiving proportionally less development resulting in less-developed highlights in relation to the mid-tones and shadows. This is the compensating effect.
Also, with many developers and compensating-style developing, the developer in the higher-density areas exhausts and makes bromide as a by-product. This bleeds over just a tiny bit into adjacent lower-density areas, making a small line of less density along the interface. Simultaneously, a bit of the more-active developer in the low-density area will bleed over into the higher-density areas, making a small line of more density on that side of the interface. The resulting double-line of high/low density between a very high and very low-density area can be quite pronounced and is called a Mackie line. It lends a look of extra definitions and sharpness in many cases and some of us strive to maximize this effect just for that characteristic look. Dilution of the developer and agitation frequency play important roles in this process.
Also, in the past I have found that a more-dilute developer and longer developing times allow me to achieve more contrast than with a more concentrated developer and extended development time. This, I believe, is due to the activity of the more-concentrated developer causing increased base fog with extended development times, which doesn't seem to be such an issue with a more dilute developer and even longer development times. For example, I could get significantly more contrast from a Tri-X negative with HC-110 at 1+63 and very long times than I could with the same developer at 1+31 and the appropriately long time for N+2 development.
Hope this helps,
Doremus