How much of a difference should there normally be, between systolic and diastolic pressures? What could cause them to be too far apart, or too close together? What other symptoms could such deviation cause?
The difference between systolic and diastolic pressures is known as the pulse pressure. (If this doesn't make sense, please see another answer of mine where I explained the meanings of the different components of blood pressure.) There is no "normal" or "should" that are well defined here. Despite that, there is quite a bit that can be said.
What causes them to be ... far apart? *
On a population level, the biggest factor is age. With increasing age, pulse pressure increases, sometimes dramatically.
Arithmetically, this is because diastolic pressure peaks at about age 55 and delines thereafter, whereas systolic pressure continues to increase throughout life (see Franklin, 1997).
Physiologically, this is because the large vessels tend to "stiffen" with age to calcification. As such, when the heart pumps blood into them (systole), they aren't especially elastic, so the diameter doesn't increase much. The result is higher systolic pressure. Similarly, in the period of diastole when flow is lower, the arteries don't rebound to a smaller diameter as robustly as young vessels, so the pressure falls. It is common in 80+ year-olds to see blood pressures like 180/60. I've never seen such a pressure in a young person.
This is the part with an obvious physiologic correlate that medical schools and board exams like to hammer on. Aortic regurgitation (a.k.a. aortic insufficiency) is the primary valvular disease associated with increased pulse pressure. That's because the incompetent aortic valve allows part of the blood to flow back into the left ventricle during diastole. If the aorta has access to the ventricle as it relaxes during diastole, the ventricle acts as a pressure sink, resulting in lower systemic pressures during that phase, increasing pulse pressure.
The converse of this is aortic stenosis, which is a valve that doesn't let adequate blood through during systole. Because the flow is less, the pumping ventricle does not change the pressure in the systemic circulation as much as it normally would.
This physiology text does a pretty good job explaining the valvular pathology in more detail.
Pulse pressure as a predictor of cardiovascular disease
There is extensive literature addressing the question of whether pulse pressure is a (semi-) independent risk factor for cardiovascular disease, beyond that provided by systolic or diastolic pressure alone. I provide a few references below. The upshot is that the relationship between pulse pressure and risk is complicated and highly age-dependent. A high pulse pressure may be a better predictor of cardiovascular events than systolic pressure itself among the elderly.
*Removed "too" because this just isn't well defined.
Franklin SS, Gustin W 4th, Wong ND, Larson MG, Weber MA, Kannel WB, Levy D. Hemodynamic patterns of age-related changes in blood pressure. The Framingham Heart Study. Circulation. 1997 Jul 1;96(1):308-15.
Vaccarino V, Berger AK, Abramson J, Black HR, Setaro JF, Davey JA, Krumholz HM. Pulse pressure and risk of cardiovascular events in the systolic hypertension in the elderly program. Am J Cardiol. 2001 Nov 1;88(9):980-6.
Franklin SS, Khan SA, Wong ND, Larson MG, Levy D. Circulation. 1999 Jul 27;100(4):354-60. Is pulse pressure useful in predicting risk for coronary heart Disease? The Framingham heart study.
Pastor-Barriuso R, Banegas JR, Damián J, Appel LJ, Guallar E. Systolic blood pressure, diastolic blood pressure, and pulse pressure: an evaluation of their joint effect on mortality. Ann Intern Med. 2003 Nov 4;139(9):731-9.