Raster GIS origins in aerial photography and remote-sensing
Raster data structures: uniform, dense arrays of values representing features require large storage capacities. Lower nominal spatial resolution/byte. Processing involves massive element-wise calculations (generally byte-by-byte) which computers are very good at. Raster output is not easily handled by conventional (e.g., Postscript) hardcopy devices.
Vector GIS origins in CAD
Vector data structures: non-uniform, sparse sets of vertices delineating features require less storage. Higher nominal spatial resolution/byte, but requires complex 2-level arc-node data structure to manage gap-and-overlay problems. Processing involves more complex data manipulations, including numerical integration (generally on floating-point data) which computers don't perform so quickly. Vector output is easily handled by conventional hardcopy devices.
The raster-vector debate is mostly moot now: run-length encoding or quad-tree algorithms provide efficient compression of many raster files (those with long sequences of uniform values). Most GIS's today now incorporate both raster and vector functions, although each has a primary structure. The choice of data structure really depends on what you're analyzing and the spatial resolution you require. Raster data structures will probably dominate in research applications over the medium term: simpler data structure, computational efficiency supports more sophisticated analyses.
3 basic GIS elements: points, lines, regions. Volumes and higher-dimension elements (e.g., volume-in-time) may be added in the future.
Points are defined by single vertices or cell (wells, cases of reportable disease).
Lines are defined by non-closed sets of vertices or strings of contiguous cells (roads, power lines).
Polygonal areas or regions are defined by closed sets of bounding vertices or clusters of cells. (Distinguish closed poly-lines from filled polygons where appropriate.)
Most GIS manipulations of spatial elements correspond to set theory: overlay, split, buffer, point-in-polygon, etc. are basically union, intersection and membership operations.
In practice, most GIS work today is really just computer-assisted cartography. Major product is hardcopy maps. People are accustomed to looking at map representations of spatial phenomena. Maps appear authoritative, provide an information "rush," are visually appealing, are best method of "selling" GIS to lay people. Do hardcopy maps really convey as much information as they purport to?
But hardcopy maps are static and dimensionally somewhat restricted, where GIS is dynamic and dimensionally unrestricted.
GIS does need selling, because it is so expensive. Appropriate levels of investment in infant GIS technologies are hard to determine. Partly due to aggressive marketing by some GIS vendors, many agencies are clearly over-invested in expensive, high-end GIS technologies they don't need or use. Facing budget cutbacks, GIS may be first to go.
The analytical sophistication of users, not the operational capabilities of the GIS, is the major bottleneck in the diffusion of GIS technologies.
GIS presents a fundamental challenge to classical statistics. We use GIS to analyze continuous, spatially-interdependent processes. But classical statistical methods were developed from analyses of discrete, independent events (draws of colored balls from urns, rolls of die, etc.). Spatial autocorrelation between proximate data elements violates classical assumptions of independence and renders classical methods inefficient or inappropriate.
This implies that "observations" of continuous spatial processes are not really "data" in the classical statistical sense. Classical statistical methods are sensitive to the number of "observations" N (degrees of freedom). But we can sample as many observations as we want from a bounded continuous spatial process (e.g., by resampling at progressively smaller raster resolutions). We can keep taking additional "observations" and boost the nominal significance of our statistical tests as high as we want. The fallacy here is that spatial interdependence implies that each additional "observation" drawn from the surface adds progressively less new information to the analysis.