In addition would require you to add more information, which you have not added here. Although it may look like you have, all you have stated is that it was a brilliant game and you got in for free (did not have to pay).

Not having to pay is not extra information about the game, it is extra information about your non payment that you did not give until an after thought at the end of the sentence. =  the moreover structure is required.

"we went to see the match yesterday in addition to having a wonderful time we did not have to pay" 

In this sentence you have added extra information onto  ""we" 

"It was a brilliant a match and additionally[in addition] it was free.

in this sentence you have added extra information about "it" 

"the match was free and in addition it was a brilliant game/match" 

in this sentence you have added extra information about the game/match both being the same thing. 

in your sentence your are talking about a game and supplying only more information about an unconnected thing your free entry.  It is not emphasising what you are saying therefore we use the moreover structure. 

It has nothing to do with informality and formality but about what you are saying, although to native speakers they would instinctively get a feel for which one to use. 

"What's more" is a better fit for a sentence about a free ball game because it is a bit less formal than "in addition/furthermore/moreover". Additionally/In addition isn't *wrong* though.

(and) what’s more

Definition:   used to add more (important) information that emphasizes what you are saying.

example:    I’ve been fortunate to find a career that I love and, what's more, I get well paid for it.

example:    Gas is a very efficient fuel. And what’s more, it’s clean.

in addition

Hello!

I'm not completely sure but I think "what's more" is typically used to add a point that is even more important, while "in addition" is used if you want to mention another point after something else. So, in your case you would like to emphasise that you didn't even pay to get in, which can be considered more important than the brilliance of the game. Thus, "what's more" would be more suitable in this context. :) I hope this might help you!