The different response is essentially that different antibodies (immunoglobulin [Ig]) are present in the blood (and associated tissues) and the mucus (and associated tissues). In the mucus membrane the primary antibody is IgA, whilst in the blood it is generally IgG.

IgA is a poor activator of the secondary immune system (the lymphocytes that do the majority of work in destroying the invader) anyway, and the cells that produce it does not get good exposure to blood-based vaccine, which means that they are not particularly sensitised to the attacker. They will still respond, but they are not as effective at it.

IgG and the cells that secrete naturally do get a good exposure to the vaccine, so the body produces many of them specifically designed to identify and attack the virus as a result of being vaccinated. This is important since in the process of doing this they signal the body to start producing the lymphocytes that actually attack the invader. [And, as an added bonus once you have had the disease or the vaccine the secondary immune system remembers the most effective type of cells to use to attack the virus and doesn't have to discover how through trial and error.]

When the coronavirus starts colonising the upper respiratory tract (particularly the nasopharynx region), it primarily encounters IgA. It is only when it essentially moves deeper into the body that it encounters IgG which then triggers the full immune response, before it can effectively colonise the lungs and enter the blood (both of which are where it causes serious health problems).

Note that once the secondary response fully awakens it attacks the virus both in the body and in the upper respiratory tract. Which is why you may start out with the same viral load when infected, but you reduce that load a lot quicker than an unvaccinated person (and thus get better sooner). [As well has having less risk of complications.]

The omicron strain has several mutations from the base strain that make it particularly good at initially colonising the upper respiratory tract. Which is why it is spreading a lot easier - an R of between 4-7 with a mean of about 5.5 has been reported (amongst a theoretical unvaccinated population, which is why the variation in results). [The original strain had an R of about 2.7.]

There are currently trials being undertaken with a nasal spray vaccine to sensitise the IgA, to get a better immediate immune response. However it is likely that these will only be available to front line medical personnel and those that can afford them, as they probably will have an effectiveness of only 3-6 months and the standard vaccines should protect against the serious side-effects of catching COVID. [We are seeing an effective six-fold reduction in the need for hospitalisation amongst those who are vaccinated.]

This is all highly simplified. Key words for more information are immunoglobin (IgA and IgG), complementary activation, lymphocytes (T-cells and B-cells), primary and secondary immune response, and memory cells.

Hope this helps.