We used to read both versions in high school French class. There was much more slang in French. Many of these were replaced by silly puns in English.
Even the names: Getafix the druid was originally Panoramix. Dogmatix the dog was Idéfix (this is actually a pretty good translation, keeping the core idea of single-mindedness, plus it has Dog in it).
The chief and bard names are the worse. Abraracourcix is a reference to someone prone to violence in French, which is why he keeps getting angry and red-faced. There’s a whole plotline about him having to go to a spa so he can lose weight and relax. Not sure why they renamed him to Vitalstatistix in English.
And the noisy bard goes from Assurancetourix (comprehensive insurance joke) to an unsubtle Cacofonix. But to answer your question, most of the bad puns were added in the English translation.
FWIW, they did a reverse butcher job with Harry Potter books. The French versions literally translated the British expressions word-for-word to the point they made no sense.
Blowing other people’s runny noses, with your mind.