Why would diffusion be faster across a non-specialised tissue?
There are several issues here:

1) Any mucous membrane is a specialized tissue for absorption.

Mucous membranes are indeed not so good for passive diffusion, that makes them absolutely perfect tools for active absorption of certain substances, almost independently from the membrane type. To provide some examples: many drugs like cocaine are inhaled and absorbed in nasal cavities, whereas the rectum is also known and favorite delivery way for medicines. 

Generally the absorption force of the mucous membrane is dependent upon how well it is vasculated, for after going through the basal membrane the substance directly enters the blood flow and quickly travels away, thus the concentration gradient on both sides of the basal membrane -- the main barrier in mucosa -- remains relatively high.

See  Bhat P. 1995. The limiting role of mucus in drug absorption: Drug permeation through mucus solution for an experimental model for this statement.

2) Mouth mucosa can absorb many substances.

There is a special term called "oral absorption" to describe the rapid drug absorption into blood flow from the mouth cavity. The mucous membrane is not specialized here, but small molecules are able to permeate here through all barriers. 

3) Advantages of oral absorption.

There are some:


The mucous membrane in the mouth cavity is very highly vascularized. The whole mouth can be seen as a bundle of skeletal muscles and every muscle requires a lot of energy and oxygen, therefore they have one of the highest vascularity rates, having much less distance between single capillaries.
Blood flow is higher in the mouth cavity walls than in other inner organs. This is mostly because muscle contractions (during chewing) lead to increased propulsion of blood through capillaries and small vessels here.
Any substance that enters blood flow here bypasses the hepatic-portal system. This means that the substance does not have to wait until it is filtered by our liver, it is immediately distributed through the whole body.
There are special reflexes from oral mucosa to inner organs. Even not so important in case of aspirin, this is very important for some placebo drugs like methyl valerate (known as "Validol" in many countries) used a lot for treating angina pectoris whose only action is to activate the cold receptors in mouth and thereby leading to reflectory dilating of cardiac vessels.




This is why many drugs, like for example loperamid, are administered only as sub-lingual tablets. And this also explains why in emergency medicine many remedies are injected directly into the tongue.