The difference between flu - and almost all other diseases, tbh - and Covid is that flu was defined by its symptoms long before we ever figured out what caused it; whereas "Covid" is defined not by symptoms by a detectable infectious agent. When people talk about "having the flu" they usually mean they were ill in the way they recognise as having flu symptoms not that they were infected with one of the influenza viruses. If you measure "flu" in the same way as we measure Covid - i.e. detectable infection with the virus - then you find that the majority of cases are characterised with mild or no symptoms.
There's some impact from mutation but Covid is nothing like Flu in that respect. Flu has an incredible propensity for recombination that produces an awful lot more variations than we see in most viruses. The main effect is that, for reasons that are unclear, immunity against Coronaviruses is just less persistent than against other diseases. Whether from vaccination or infection, the protective effect tends to decline more rapidly than expected.
Incidentally, even for the things that the UK only vaccinates against in childhood there is a decline in protection over time. Some countries, Germany included, give you boosters against Measles and so on every ten years to keep your protection up.