Hi Lesley,
Please don't take this personally, but I will 'super'-clarify a few points regarding
your question
If I were a pedant, I might say the advantage comes when you need to PP (post process), rather than when shooting - in fact there are a couple of minor disadvantages when shooting; the larger files take longer to write to the memory card and fill it quicker - sometimes this is important.
Equally, the main advantage comes when you need to do a lot of PP, rather than just display the image on a website or just print, for both these tasks, jpg is perfectly adequate.
However, you can fix a lot more issues in PP AND end up with a higher quality result if you shoot RAW.
I don't know how old you are, but back in the film days, if you had a good 6" x 4" snap and the negative, a 24" x 16" enlargement made from the negative should always have been better quality than one made from the print.
So the simple answer to your question is YES
By all means have another go at PP - I assume you didn't save over the original file - and repost so we can see the improvement.
Cheers,