Sorry if this is an ignorant question. But I just can't fathom what the advantages are of using a differential pair for signals not leaving the PCB. I mean, the usual argument for using a differential pair is to increase common mode noise immunity, which makes a lot of sense in cables and connections off the board due to all kinds of common mode noise that can exist in such environments. I just don't see the big advantage if the signal begins and ends on the same board. For instance, noise from adjacent traces couples asymmetrically to the pair, and one can probably say the same for broadside coupled traces. This obviously introduces noise in the differential signal. So why not just use a single ended signal and then increase the minimum distance to other traces if one wants less crosstalk?
A trace on a board may also be subject to radiated signals coming from elsewhere. A differential pair will likely pick up less cross talk from such sources.
A board may be part of a system that is subject to regulations that limit the radiation it is allowed to produce. A differential pair will likely produce less emissions.
A differential receiver will be less impacted by the receiver's reference level shifting due to power supply noise or ground voltage variation.
There are applications where differential signaling cannot be avoided. For example, when you need to connect a Gigasample ADC (analog to digital converter) to a processor. Gigabit links cannot be implemented without low-voltage signal swing (200-400mV), and the low-voltage signal cannot be transmitted over single-ended traces due to crosstalk, as already explained in other answers. That's why the signaling schemes as LVDS (Low Voltage Differential Signaling) were invented.
If you have embedded USB3.0 devices or any SATA or PCIexpress sockets, differential signal tracing is a must. Most obvious example would be DDR3/DDR4 embedded (soldered down) memory. In all cases the traces do not leave PCB.
There are a number of reasons why differential pairs are helpful on PCBs even when the signals don't leave the boards. The main benefit is that they are less susceptible to crosstalk. You say that "noise form adjacent traces couples asymmetrically to the pair" which to a very slight extent is true, but if the differential pairs are run together the signal is coupled into both nearly equally. There will be a slight difference but for all practical purposes the noise is coupled into both signals identically. Generally speaking, differential signal routing provides twice the noise immunity as a single-ended routing.
Signal integrity and EMC covers a broad spectrum and dynamic range. Without details of the SNR requirements , CM E-H field strength, any imbalance in CM impedance gets translated into differential noise.
In some cases timing skew is critical as well as path length in 100 ps increments or less for CML and PECL and LVDS signals.
For RF, and logic controlled impedance is critical and paired tracks are often used , even with differential video signals.
In the end it's all about SNR whether it is logic or analog as even high speed logic follows analog rules.