Apparently these are both translations for older brother and older sister.
What is the difference?
Grand frère and grande sœur, literally "big brother/sister", is the casual way to name an older brother or sister while petit frère and petite sœur (small brother/sister) mean younger brother/sister.
There are also specific and more literary names to define siblings order.
Aîné(e) etymologically "born earlier" (Old French ainz né from low Latin *antius natus): means either the first-born → l'aîné, or older → mon/son/ton/leur aîné(e).
Puîné(e) etymologically "born later": any child but the first-born. Le puîné is the second born. This word is no longer used in modern French, but survived as "puisne" in English, as Frank wrote.
Cadet(te) originally Capdet from low Latin capitettus → Little head/chief; Captain, is a synonymous of puîné. The reason why is that captains were often non heirs from Gascogne aristocratic families (D'Artagnan and the three musketeers are cadets de Gascogne). Cadet gave the English "caddie". When we say le cadet in modern French, it always means the second-born while son cadet means "his/her younger brother". Note that in Gascogne, like in many other southern provinces, the aîné was not necessarily the first born but the son chosen by the father to be his heir. Consequently a cadet could well be the first-born.
Benjamin(e), originally both the favorite and younger child of Jacob in the Bible, used in French to name the last-born child.
It's a difference in language level, like in English between big bro and older brother. Grand is more colloquial. Ainé is more formal, and in fact significantly more so than English's older brother would be (maybe it's even closer to elder brother).
I think aîné's "first" meaning in French is first born (although coming from the Low Latin antius natus, where antius is only the comparative, not superlative, of ante) , which is not necessarily the case for grand frère/soeur, which would just mean older than. CNRTL gives: Fils aîné, fille aînée. Né(e) le premier (la première), le plus âgé ou la plus âgée par rapport aux autres enfants de la même famille. Synon. premier-né. Anton. cadet, puîné, second.
In an expression like:
where you are introducing your children, it is certainly your first born that you are introducing.
However, in the expression:
Je suis son aînée de cinq ans.
aînée just means older than again.
Aîné has yet another meaning in, e.g.:
Des activités sportives pour les aînés.
Where it means older folks.
So we have three possible meanings for aîné, whereas grand frère/soeur just means older sibling. Note also that there is a cultural baggage associated to aîné which biases the meaning towards first born, as historically, le fils aîné would get preferential treatment over other brothers/sisters (typically getting a better education and the most promising career).
Note also that grand frère/soeur are in a more relaxed language register.
The opposite of aîné in the sense of first born would be: puîné (lit.: born next) or cadet. Puisne is attested as a legal term of art in British English to mean "inferior in rank" (the aîné getting all the attention, better education, better position...), but aîné doesn't seem to have made it to English.