Why should collected blood from patients be analyzed as soon as possible?
Biological samples, including whole blood, are not at equilibrium. Biological processes, e.g., cellular metabolism, continue. Cells interact with their environment and a tube is not the same thing as the human vascular system. For one, in humans, on average, the entire volume of blood recirculates once every minute (see Costanzo Physiology Ch 4). Recirculation, with the associated access to nutrients, waste drop off, and exposure to various different compartments doesn't happen in a specimen tube. Nutrients are consumed, waste collects, cellular stress occurs, intracellular components leak out, cellular morphology changes... You can slow this down by putting samples on ice and collecting blood in a tube with components that preserve the sample for the specific test. There's a nice discussion of this in Lange's Guide to Diagnostic Tests, Chapter 1. 

Given ongoing biological process in a very different environment, you can understand how interpretation of a particular test result depends on consistent collection and storage methods. Different tests are going to be more or less sensitive to time since collection and storage methods. There's a good deal of data on this for whole blood in this review.