In general with rail tickets in the UK it is not so much about where you buy them as when you buy them and what restrictions you are prepared to have.
"advance" tickets can be significantly cheaper than "off peak" or "anytime" tickets but they must be bought in advance and they lock you into a specific train. They are also not available on all routes and are limited in number. Sometimes there can be multiple different advance fares for the same train with the cheapest ones selling out first.
"super off peak", "off peak" or "anytime" tickets will be the same price whenever you buy them and they aren't limited in number.
On some origin/destination pairs "super off peak", "off-peak" and "anytime" tickets may be further split by "route". A "route" may be an actual physical route or it may be an operator. Tickets limited to one route or operator can be a bit cheaper than tickets with route "any permitted" but are obviously less flexible.
"advance" tickets are always singles. "off peak", "anytime" and "super off peak" tickets can be either singles or returns. Return tickets can be either limited to the same day (e.g. "off-peak day return") or valid for a longer period (e.g. "off-peak return"). Sometimes, especially with off-peak tickets a return is only marginally more expensive than a single. Unfortunately on many shorter journeys "off-peak return" tickets are not available.
Note also that on routes where both are available "off-peak day return" tickets often have stricter time restrictions than "off-peak return" tickets. I have no idea what the time restrictions of off-peak singles are like.
There are both first class and standard class fares. Generally "off-peak" fares don't exist for first class. First class fares are generally substantially more expensive than standard class though occasionally the cheapest available fare for a journey will be a first class advance fare because all the standard class advance fares have sold out.
If you look carefully on the websites they will usually tell you what exactly they are selling and let you choose tickets other than the cheapest available but they don't like to make it too obvious.
It is difficult to predict whether your plane will be on time or how long it will take to get through immigration and customs. So if you buy an advance ticket you will need to have substantial padding in your schedule. Personally I rarely buy advance tickets because I do not consider the cost saving to be worth the inflexibility and the extra waiting around.
Be careful when buying online, some sites (notably the trainline) will charge extra for "booking fees" or "delivery". These charges will be on top of the price for the ticket itself.
You can find the full list of possible tickets for your station pair at http://www.brfares.com/#!fares?orig=GTW&dest=BTN note that this site includes fares that are not available for the general public in addition to the regular publicly available fares.
It looks like advance tickets do exist but only for "southern" who operate a minority of the services on the route. If you are prepared to book in advance and add a lot of padding to your schedule to make sure you don't miss your train then this will be the cheapest option but I don't think sitting around for an extra hour or so is worth saving a fiver.
Given this is a journey from the airport I will assume you are not planing to return the same day. I'm not sure if you plan to return at all. I will also assume that you intend to travel standard class.
There are no "off peak single" or "off peak return" tickets on this route. There are some "anytime return" tickets but they cost the same as the equivalent pair of "anytime day singles". There are some "off peak day return" tickets but they are only worth buying if you actually intend to return on the same day.
So I would say your best bet is to wait until you get to the station, look at what trains are actually available then buy the most appropriate single ticket based on the train you actually plan to catch.
- If you plan to catch a "southern" or "gatwick express" train then you want the "anytime day single" route "any permitted". Cost £10.20
- If you plan to catch a "thameslink" train on a weekday then you want the "anytime day single" route "thameslink only". Cost £8.50
- If you plan to catch a "thameslink" train on a Saturday, Sunday or Bank holiday then you want the "super offpeak single" route "thameslink only". Cost £6.50