The assertion is the same — You have a driving license —, since you can’t have it without having got it, but the focus is different :

• « J’ai eu mon permis de conduire » puts emphasis on the obtainment of the driving license
• « J’ai mon permis de conduire » puts emphasis on the actual possession of driving license

For instance, you would answer « J’ai eu mon permis de conduire » if you were asked something like “Have you passed any examination?” and « J’ai mon permis de conduire » to “Are you in possession of any official document?”.

Take the verb “danser”, “to dance”. Its “indicatif présent” is “je danse” (“I dance” or ”I’m dancing”); its indicatif passé composé is “j’ai dansé” (“I danced”). “Avoir” appears here as “ai”, but the main verb still is “danser”.

In your case, the word “eu” is the past participle (singular masculine) of the verb “avoir”. Hence, ”j’ai” is the indicatif présent of “avoir” and “j’ai eu” is its indicatif passé composé. In “j’ai eu”, “ai” is “avoir” used as an auxiliary verb. It’s only by chance that “avoir” also is the main verb.

So, if the sentence were translated with “avoir” with its meaning of “to possess”:

J’ai mon permis de conduire. I have my driving license.
J’ai eu mon permis de conduire. I had my driving license. (I don’t have it anymore.)

However, “avoir” has lots of different meanings, one of which is “to obtain, to acquire“. And, without any contest, this meaning is much more likely in “J’ai eu mon permis de conduire”.

J’ai eu mon permis de conduire. I got my driving license.

So “J’ai mon permis de conduire” puts the emphasis on the fact that you have your driving license; “J’ai eu mon permis de conduire” puts the emphasis on the fact the you obtained it.