Pros
Some interesting people working here. Generally staff are passionate about history and heritage, and this is probably one of the few places you can put that History or Fine Arts degree to practical use; Flexible attitude to leave, sick leave, lateness etc; New London office is nice and modern with pleasing decor and ambience; Get to make a contribution to preserving the country's history and culture, which I'm sure all but the crudest of philistines would support. Depending on your role you could be working on some very high-profile, interesting cases; Efforts to shift away from what is still a very traditional, white middle-class organisation. Though easy to be cynical about, this does seem to be a sincere move and not just politically correct PR nonsense (though note below); Opportunities to travel across the country (dependent on role); Free access to English Heritage sites (though this perk may have been phased out now since the recent HE/EH split).
Cons
As you might expect, quite a traditional public sector environment, with all that that entails. Those looking for a high-octane, edge-of-your-seat company to work for are best off looking elsewhere. Some staff very stuck in their ways, and some of the more academic types can be quite arrogant and big-headed; Abysmal IT which as often as not hinders productivity rather than improving it; Some customer-facing roles (e.g. listed buildings or planning teams) may involve dealing with some very angry, ill-mannered people; However the main con of HE would have to be the appallingly low wages, due to HE being a government-funded body. Cuts to the organisation since the austerity era kicked in means that wages have scarcely risen in that time. Now, no one goes into the heritage sector to make big bucks, but quite simply the salaries are not enough to live on in London (or probably one of the other more expensive locations such as Cambridge or Bristol). It is very common for HE employees to be dependent on their partner's income, with the other half doing the 'real' job which earns a decent salary. Management are aware of the wage issue but in many ways their hands are tied by government paymasters (culture and heritage always being the lowest priority of course). HE is trying hard to attract people from beyond their traditional white middle-class demographic, however the poor salaries on offer directly conflicts with this aim. It is a real shame as many talented, enthusiastic people from less well-off backgrounds end up leaving for better-paid positions elsewhere; Allied to the above would be the poor opportunities for professional growth. Staff cuts means there's little on offer, and when vacancies do arise competition can be fierce. The promotion process is not very transparent and favouritism clearly plays a role in some appointments. Some roles can be quite repetitive and offer little variety or opportunity to learn new skills.