What problems might a single ground rod above 25 ohms resistance cause?
Well it is the National Electrical Code that requires 25 Ohms or less. Notice the exception after #5.
Here:

250.53(A)(2) Supplemental Electrode Required. A single rod, pipe, or plate electrode shall be supplemented by an additional electrode of a type specified in 250.52(A)(2) through (A)(8). The supplemental electrode shall be permitted to be bonded to one of the following:
(1) Rod, pipe, or plate electrode
(2) Grounding electrode conductor
(3) Grounded service-entrance conductor
(4) Nonflexible grounded service raceway
(5) Any grounded service enclosure
Exception: If a single rod, pipe, or plate grounding electrode has a resistance to earth of 25 ohms or less, the supplemental electrode shall not be required.
(3) Supplemental Electrode. If multiple rod, pipe, or plate electrodes are installed to meet the requirements of this section, they shall not be less than 1.8 m (6 ft) apart.
Informational Note: The paralleling efficiency of rods is increased by spacing them twice the length of the longest rod.
(B) Electrode Spacing. Where more than one of the elec-trodes of the type specified in 250.52(A)(5) or (A)(7) are used, each electrode of one grounding system (including that used for strike termination devices) shall not be less than 1.83 m (6 ft) from any other electrode of another grounding system. Two or more grounding electrodes that are bonded together shall be considered a single grounding electrode system.

The reason contractors elect to install a second ground rod is that it is cheaper than paying for the ground resistance testing.
The advantage to a low resistance is during a lightning strike or large ground fault even a small resistance creates a large voltage differential. This could be dangerous for persons in contact with grounded parts of the system at that moment. Hence the reason for the requirement.
Otherwise, during normal operation there isn't a noticeable advantage to consumers.
For an older home you may already have a grounded metal water pipe exiting the building so you already have a supplemental electrode. If not adding an extra rod may help protect you and your valuable electronic equipment during a lightning strike or large voltage transient on the line.
Good luck!