The resistor functions as a "constant obstacle" to the current flow - the more "force" (voltage) you apply across the resistor, the more current flow you'll get.
The above description is not the most accurate one because it neglects the dependency of resistivity on temperature and frequency, electromigration and etc., but it is good enough for intuitive explanation.
Diodes, on the other hand, are not "constant obstacles". There is a region inside diodes which is called a depletion region (or, sometimes, space-charge region) which is the most interesting part of a diode - it is this region which makes diodes (and transistors) so different from resistors. Depletion region (and effects associated with it) is very sensitive to the externally applied voltage.
For regular PN diodes, when the forward voltage across the diode increases, depletion region shrinks in size and allows for the current to flow "more easily" (when depletion region shrinks the voltage across it decreases). You can think of it as "doubling" the effect of the voltage: it causes more current to flow, and causes depletion region to shrink which allows for even more current to flow.
When reverse voltage is applied to the diode, depletion region expands. It is not causing the current to become smaller (at zero voltage there is no current through diode, but there is some current when the diode is reverse biased), but the expansion increases the voltage across the depletion region thus "neutralizing" the externally applied reverse bias, which prevents from current flow to ramp up.
The regular PN diode is based on the two points above: it allows for high currents to flow under relatively constant voltage while it is forward biased, and prevents reverse currents while it is reverse biased.
Zener (and Avalanche) diodes are similar to the regular PN diode, but they exploit the reverse-bias as the operating mode. As we said, the current through reverse biased PN diode is negligible because depletion region "neutralizes" the externally applied voltage. However, everything has its limits: when the reverse-bias of the diode crosses some threshold, the depletion region can undergo Breakdown (usually, it is non-destructive effect). When Breakdown occurs, it is like the depletion region is not there at all, and the current ramps instantly to very high values, while the voltage across the diode remains essentially as it was a moment before the breakdown.
Many people will want to crucify me because what I wrote. Yes, this is very simplified and inaccurate description of PN (Zener, Avalanche) diode's principle of operation. However, I believe that this model allows for intuitive understanding of PN junction, without any preliminary knowledge in semiconductors.