I've visited both many times and both are great. The beach and water is relatively the same (fluffy white sand, emerald green waters) and both have good places to eat. Siesta Key is obviously close to Tampa / Clearwater, so if you want some big city things to do, that may be a better fit (i.e., NFL game, MLB game, etc.). Offshore / Inshore fishing is roughly equivalent - both have buried wrecks and structure all over - but I would say the intracoastal in Siesta Key is more vast and interesting to explore than the Destin area. Not sure where you are driving from - but Destin is a shorter drive from TN than Siesta Key - probably 2-3 hours shorter drive depending on your origination. Destin traffic is a mess, but I've also experienced bad traffic in Siesta Key.
Long story short, both are great and you won't be disappointed in the beaches. Not sure how they compare on price - seems like Siesta Key might have been slightly cheaper, but honestly haven't compared in a long time. I would like slightly East or West of Destin if you decide to go to that area - less crowded and same amenities.