Ozempic/Wegovy are brand names for a medication called semaglutide. It's called Ozempic if its prescribed for diabetes, and Wegovy if it is prescribed for weight loss. It works by mimicking a natural hormone called GLP-1, which has a number of effects on the body. GLP-1 appears to decrease hunger, increase feelings of fullness, increase the amount of insulin the pancreas produces and may decrease cravings (jury still out on that one).
Mounjaro/Zepbound are brand names for a medication called tirzepatide. Again - Mounjaro if it is for diabetes, Zepbound if it's for weight loss. The reason why patients looking for weight loss are prescribed Mounjaro in the UK is simply because Eli Lilly have not launched tirzepatide with the brand name Zepbound in the UK yet - it's only in the US. They are furiously buying up manufacturing, pen making and printing facilities so that they can roll out Zepbound in the near future. But don't misunderstand me here, the medications are exactly the same. It's like Opal Fruits and Starburst - only the name is different.
Similar to semaglutide, tirzepatide mimics GLP-1 in your body. However, in addition, it mimics a second hormone called GIP, which also helps in regulating blood sugar. It appears that the combination of both mimicked hormones does two things a) makes you lose a bit more weight than semaglutide and b) it may reduce the side effects like nausea and constipation.
As to which one you should pick, that's a difficult one. Tirzepatide seems to be the obvious answer. However, the combination hormone drug is relatively new. Semaglutide has been taken by diabetics since 2018. Semaglutide also had a very similar sister drug called liraglutide (marketed as Victoza and Saxenda) which has been prescribed since the mid-2000s. So, at least in the medium term, there doesn't appear to be any unknown effects of semaglutide. I.e. if the medication made you grow horns, the diabetics would have horns already. There's a little more "unknown unknowns" about Tirzepatide.
That, and you have to take into account supply. Supply of semaglutide seems to be getting better - probably because people are switching or starting on tirzepatide. The US FDA have stated that there will be sharp shortages of tirzepatide in the second half of this year.
All that said - if both semaglutide and tirzepatide were offered to me I'd probably go for the latter. If I am going to be injecting myself with a very expensive drug for months/years, I want the one that actually does what it says on the tin. Unfortunately, as I live in Ireland, tirzepatide won't be available for some time to come.