I worked in a job centre for 14 years, and for the first 12 I loved it. Yeah, there are some shit staff who wouldn't last two minutes in the private sector but have been poorly managed so get away with doing fuck all. Some of the managers are "yes people" and so promotion on temporary detached applications can be a popularity contest. However, in my opinion and my district, the permanent jobs tended to go to people who were better suited to the role.
The managers I had were really supportive of mental health issues which was great but sadly did mean some people took the piss. Likewise, it's v supportive for parents and working patterns, time off etc which again did lead to some people taking the piss.
The customers were everything from the loveliest people who were going through a terrible time and that you really got a sense of fulfillment from helping, to vile human beings who would make your skin crawl (paedophiles, rapists etc)
Sadly over the last year or so in my office, a change in higher management made the office a horrible place to work. Loads of my colleagues have left, including me. I would have been happy to stay there longer but the district manager and his sycophants ruined it and it actually makes me sad cause my office was such a good place to work and the majority of us always supported each other.
I am now in a different CS department and it is awesome. My manager is supportive and is really keen on helping staff develop. She wants to overhaul the section to get rid of the bureaucracy (as much as possible) and streamline things. She's willing to tackle poor performance but is open and available if anyone needs her help. I actually look forward to work in the morning and am excited about what is coming up.
Sorry, this is a really long post! Basically, TLDR it totally depends on department and management. You will always have benefits such as flexi, pension, ability to move departments, but other than that, it can be the most rewarding job or the most frustrating.