The senior food is often lower in certain nutrients such as sodium and higher in other nutrients to help maintain healthy organs in an ageing cat. The differences are subtle on paper but they do make a big difference to an older cat as their needs (as with humans) differ as they age and their organs start to become more susceptible to certain conditions such as kidney disease.
You're unlikely to notice much difference in the health of your cat by switching to senior but, as has been said, it's a preventative measure - by switching to senior you're trying to keep your cat healthy for as long as possible.
The vets don't always know best when it comes to the intricacies of diet but all vets understand the basics of nutrition (as you'd expect!) and the recommendation to switch an older cat to a senior diet is a good one
As has also been mentioned, in older cats it is good to get bloodwork done semi-regularly too as they become at risk of certain diseases and bloods will pick them up in the early stages. These are only advisable and that's why your vet didn't push the matter when you said you couldn't afford it; they're not vital as they're just to rule out any issues. £200 does sound a lot though, not sure what your vet was planning on testing for that price!
Anyway.. I wouldn't doubt your vet too much - it sounds like his biggest flaw is not explaining his logic to you properly. Vets often forget that owners need explanations as we're not all fully trained in veterinary medicine like they are. If you have questions you should ask your vet - he should be able to answer any and all questions easily and if he can't give you satisfactory answers only then should you be doubting him
