I think having the second child really means more involvement from the other partner. With the first, you also don't really relax, as you are afraid of everything, but with the second, you are not going to rush to the emergency room if they knock their head straight away, as you have been through it all before. I have found with all the people I know, that the second kid is tougher, physically and mentally, than the first, as you do tend to molly coddle the first more. I'll also say the second will be more independent than the first.
If you/partner is nursing, then the other one will need attention, be also aware that no matter what you do, the first will be jealous of the second's attention, and in time, the second with the first. They just do.
My wife and I talked about this the other day, whilst on holiday and we both agreed the step to have a baby was a real smack in the face, and real wake up call and you can never explain to someone just how hard it is having a baby for the first time. We both also think the step to have #2 was equally hard, but from a different perspective e.g. I no longer could work when she was nursing, so had to get some time to do this elsewhere. We'd also say it was more relaxing, as you are aware of what is going on, which you just aren't for a long time with just one child. However, and delightfully, we've found the step to #3 much easier. The kids adore #3, and the eldest is always helping fetch, reads to her, plays with her and generally is a benefit to have around her, and we feel this will only improve with time.
So, in short, the step to #2 is just as hard, but in different ways. It's also more relaxing.