Advance fares are only valid on a specific train. The other two fare types that I can see for this route - Off-Peak and Anytime - are flexible fares, i.e., they are not tied to a specific train. "Off-peak" means you can travel on any train outside the rush hour, "Any time" means you can travel on absolutely any train at all. (Just noticed you're arriving on a Sunday... off-peak tickets are valid any time on Sunday)

So... you're not going to take any risks by buying your tickets in advance, but on the other hand you're not really going to gain anything because you will have to go to a ticket machine to collect your ticket and there isn't that much difference, time-wise, between punching buttons on the machine to collect a ticket and punching buttons on the machine to buy a ticket from scratch.

As for getting to St Pancras... while the Heathrow Connect is slightly quicker you may want to consider using the Piccadilly Line as you wouldn't have to change trains that way.

There is always a lot of engineering work going on on the railway tracks on Sundays. If your route is affected then your journey may take a lot longer than you had hoped, and some of it may be on one of those dreaded "rail replacement buses" (some more dreaded than others... some rail companies hire in urban double decker buses and use them for journeys that they are simply not designed for)

PS Where are my manners? Welcome to FT!