Reality is that these are expensive errors that can be avoided by systematic planning but made far too easy by default software options. I've seen weeks/hundreds of thousands of dollars wasted by proceeding with data analysis without properly considering the coordinate reference system.
If spatial accuracy is a concern and
- your data is vector-based lines or a polygons, there is no guarantee that the edges between your vertices will follow the same trajectory in a different coordinate system. This means that the spatial relationship (intersection, overlap, etc) between two geographic objects depends on the coordinate system and the relative density of points.
- your data is raster-based, you need to resample pixels values to translate between coordinate systems. If your pixels represent absolute/measured values (eg population), you need to consider this carefully.
- In both vector and raster, you need to deal with dateline and pole issues. The earth is not a cartesian plane.
- In both cases, you also need to consider more than the projection. You also deal with datums which define the relationship between the actual earth surface and the theoretical sphere/spheroid the projections are based on. Get this wrong and your results might "look" OK but be meters off - you might not see it visually but it will cause problems for high-precision analysis.