"World-famous" vs "internationally recognized"

I am writing a motivational letter for a university application and I try to decide if I should use the expression world-famous or internationally recognized in a sentence:

...coming from family of internationally recognized doctors and artists...

While I am rather proud of my ancestors I would not want to sound arrogant, and I'm afraid that I don't understand the difference between the expressions mentioned before. Could someone give me a little explanation?

Internationally recognised is a bit more modest than world-famous in that recognition is milder than fame.

At the same time it's a more significant boast in that we tend only to use recognised about people who are commendable. We would say that Al Capone is world-famous, but not that he's internationally recognised. This is related to the fact that recognised also tends to be used more of the respect someone has in their field, while fame of the more general degree of recognition. There's an implication that people recognised in their field are so because they deserve to be and are judged as such by their peers, while fame is more fickle.

This also means that one can claim to be recognised oneself (or make such a claim about one close to oneself) more reasonably; there are hopefully reasonably objective criteria by which one can make such a claim (awards, invites to speak, professional qualifications) than a claim to fame.

In all, recognised serves better than famous here.

I wouldn't mention my family. In my opinion it does sound arrogant, no matter how you say it. Instead, I would let my C.V. and bibliography speak for themselves.

IMO, you can keep it simple and yet make a point.

.....family of acclaimed doctors and artists...

....family of distinguished doctors and artists...

(publicly acknowledged as excellent)

Reference: Collins dictionary.

There is some crucial context missing in the question: In which country are you applying? In most English speaking countries, both versions would be way out of line when applying to a renowned university and just plain ridiculous when applying to an insignificant one. (Things may be different in chaotic countries with lots of corruption, so I am going to assume you are applying to a North American, Australian or Central/West European country or one elsewhere with a similar cultural and economic situation.)

I am not claiming that your family background won't play a role in the decision process. If you have a rare last name associated to world famous or filthy rich people, your chances will probably be improved. Though if you make it blatantly clear that you are relying on this advantage you are probably going to ruin this effect. (Except for the 'filthy rich' part. Here a concrete description of how you would bribe them might help in some cases, I suppose.)

If people are not going to recognise you as a member of that family or can't even be expected to know the family, the only thing I can imagine might have a positive effect is subtle hints or casual mentions in the context of something that is actually relevant to your motivation:

"My parents raised me with the goal that I should some day follow the family tradition started by medical scientist Henry Jekyll and horticulturalist Gertrude Jekyll, and become a successful doctor or artist. As my school grades show, I have had other things in mind so far. But at this point in my life I have resolved to actually surpass my wider family's expectations. My ultimate goal would be to devise a genetically engineered organism that causes all the world's rainforests to be replaced by tennis lawns, and find sound artistic justification for actually carrying this through. (Funding will not be an issue.) - Kevin Beau Hyde"