Intutive difference between Hidden Markov Models and Conditional Random Field?

I understand that HMM are generative models, and CRF are discriminative models. I also understand how CRFs' are designed and used. What I do not understand is how they are different from HMMs'? I read that in the case of HMM, we can only model our next state on the previous node, current node, and transition probability, but in the case of CRFs' we can do this, and can connect an arbitrary number of nodes together to form dependencies or contexts? Am I correct here?