So, when does it make sense to have an FA? I have one now.
Nords brought up the key issue -- the fee. If you have $500,000 with a FA that charges a 1% fee, you are paying $5,000 for whatever service they are providing in addition to any fees for the mutual funds they put you into. Ask yourself, are you getting value for that fee?
The other approach is a fee only planner that may run a few thousand once a year or more often in some cases. There you get some advice but not the "active management" you get from the guy mentioned above. You'll have to rebalance your portfolio all by yourself. The stress of doing that obviously gets to some people.
You can save all the fees by doing some educating of yourself. There's a reading list for getting up to speed (do a search or ask for help). It's probably the best paying and easiest second "j*b" you'll ever find. If you are a basic passive investor, it might take you 30 minutes every year or so once you set up your asset allocation. Where else can you make $10,000 per hour?
I've described why you don't want one but I do realize you asked why you would want one. Here, I'll say you would want one if you have a very complicated financial life with many conflicting demands. A good FA can steer you among many of the issues (insurance, business planning, wills, trusts, and - yikes- even annuities). You would also need be have a substantial amount of personal wealth. I doubt that 1/2% of the the people in the US meet that level.
If you are in this level, you would want a fee only planner that doesn't have a big interest in selling you things. You would have to interview carefully for who to hire. You would still need a lawyer and probably a CPA. Both of whom would probably tell you that you don't need the FA.
Of course, my advice to you if you have a very complicated financial life is to simplify your life for your own peace of mind. Then you don't need a FA.