TLDR: They deal with similar, but different, notions of "chirality".
First, the definitions, from the IUPAC Gold Book:
asymmetric carbon atom
The traditional name (van't Hoff) for a carbon atom that is attached to four different entities (atoms or groups), e.g. Cabcd.
An atom holding a set of ligands in a spatial arrangement which is not superposable on its mirror image. A chirality centre is thus a generalized extension of the concept of the asymmetric carbon atom to central atoms of any element, for example N+abcd, Pabc as well as Cabcd.
The geometric property of a rigid object (or spatial arrangement of points or atoms) of being non-superposable on its mirror image [...]
Having the property of chirality. As applied to a molecule the term has been used differently by different workers. Some apply it exclusively to the whole molecule, whereas others apply it to parts of a molecule. For example, according to the latter view, a meso-compound is considered to be composed of two chiral parts of opposite chirality sense; this usage is to be discouraged. [...]
From the first and second, we see that the idea of a "chiral carbon" is unnecessarily restrictive, because other elements such as nitrogen or phosphorus can also have four different substituents (for elements in Period 3 and higher, this includes lone pairs). This isn't the point of your question, but is worth mentioning.
From the second and third, we see that the idea of a "chiral centre" is related to, but is not the same as, the idea of "chirality". The "chiral centre" refers to chirality in the local environment of the atom (it says that the atom is not superimposable on its own mirror image). On the other hand, the term "chirality" is almost exclusively used to refer to an object as a whole, i.e. the molecule and not just one particular atom. In fact, IUPAC discourages the usage of the word "chiral" in the local sense, as seen in the fourth definition.
The idea of "local chirality" (as defined by a "chiral centre") is useful, but one should not read too much into it. The presence of "local chirality" about a carbon atom (for example) can be a quick and easy guide to determining whether the molecule as a whole is chiral, and no doubt you will have seen examples of these. Why does this work? Very loosely speaking, if the "local chirality" isn't "cancelled out" by any other factor, then the molecule as a whole has some net chirality. More formally, if the molecule is to be achiral i.e. superimposable on its own mirror image, then the "local chirality" must be reflected onto something that is its own enantiomer (since it is locally chiral, it can't be reflected onto itself).
At the same time, a compound can have chiral centres but not be chiral: this is the case with meso compounds like meso-tartaric acid. Here there is local chirality at C-2 and C-3, but in a sense it "cancels out": C-2 reflects onto C-3 and vice versa. The opposite is true, too: a compound can be chiral but not have any chiral centres, like an allene.