How similar are Circulating Tumor Cells and Cancer Stem Cells?

There are several competing models of metastasis, and this question does go right to the differences between them.

The primary thing to remember about CSCs is that all evidence suggests that they are a tiny, tiny subset of tumor cells.

CTCs, meanwhile, consist of whichever cells manage to acquire the right combination of motility, invasiveness, and resistance to anoikis (apoptosis caused by lack of attachment to neighboring cells or extracellular matrix).

So, one possible explanation is that CTCs are drawn from the whole population of tumor cells, and that CSCs therefore make up a very small, nigh-undetectable subset of CTCs-- but that this is the subset that ends up founding the distant metastases while the rest of the CTCs get cleaned up by the immune system or lie dormant in the target tissue. You could call this the stochastic model: CSCs form some small, randomly determined fraction of the CTCs, but they preferentially survive and found secondary tumors.

Another model posits that the cells undergo substantial changes in phenotype driven by responses to their microenvironment. CSCs, like normal stem cells, presumably require a pretty specific niche to maintain their stemness. CTCs, meanwhile, acquire their invasiveness and anoikis-resistance in response to conditions in the tumor, such as hypoxia. So, another possiblity is that, in response to stimulus from the microenvironment, a cell with the potential to be a CSC undergoes some phenotypic changes and becomes a CTC. It circulates for a while, finds a target tissue, extravasates, invades the target microenvironment, then starts re-setting up the microenvironmental niche to re-establish its stem-like properties. Call this the dynamic hypothesis: it's a single potential CSC the whole time, but it looks like a CTC while it's circulating and like a CSC only in its niche.