They aren't the same; the fares structure for this route is very confusing and unhelpful.
The ticket described as 'valid only on Thameslink' is issued to London Terminals. This means it is valid for travel as far as City Thameslink when coming from the south, as is the case here. Govia Thameslink Railway (GTR) is the company that operates Thameslink, Southern, Gatwick Express and Great Northern branded trains; their position is that "Thameslink only" tickets are not valid on their Southern or Gatwick Express brands.
However, it has long been thought that this is incorrect for various legal reasons and there is an ongoing
class action lawsuit regarding Penalty Fares and additional fares that GTR has charged to passengers using "Thameslink only" tickets. Suffice it to say, if you want a hassle-free journey then stick to Thameslink-branded services, but if you are happy to potentially appeal a Penalty Fare (in our experience, this is likely to be successful) then you could use any brand of GTR services to any station that falls within the definition of London Terminals when coming from this direction, i.e. Victoria, London Bridge, Blackfriars or City Thameslink.
The ticket somewhat misleadingly described by Trainline as 'any off-peak train' is actually routed 'Not Underground' and is issued to London Thameslink - this means it is valid for travel as far as St Pancras. Since the route only excludes the use of Underground trains (which such a ticket wouldn't normally be valid on anyway) it is perfectly valid on all operators' services (including all GTR brands) and you should not face any difficulties in doing so. The restriction is more one of routing, since there has recently been a negative easement introduced which prohibits travelling via Victoria or Waterloo to London Bridge etc. on a London Thameslink ticket (something that was previously permissible and which you could get itineraries for).
So, in short, I'd say the one described as 'valid on any off-peak train' is the one you want to buy, as it affords you greater flexibility for the same price. It allows you to use other GTR brands as far as Gatwick Airport or East Croydon without hassle, and it also allows you to continue onto Farringdon or St Pancras should you wish to do so.
If you want the flexibility to use
any operator or route to any London station, you're best off buying a ticket to somewhere like West Hampstead Thameslink. This is then routed "+Any Permitted" and not only avoids any potential difficulties with brands or routes, but also includes a return journey on the Underground between two recognised National Rail <> Underground interchange stations (see the
list about halfway down this page). This does cost slightly more at £27.70 for an equivalent Off-Peak Day Return, but you might consider that a worthwhile tradeoff - especially if you can make use of the Underground validity.
