Regarding cytotoxicity, it is well said that 'the dose makes the poison' thus anything can be cytotoxic at a certain concentration, mechanisms can differ. The consequence of poison is the same for all and predictable, death - at the cellular level here.
Mutagenicity is a concern at sub-lethal doses and hence of great concern because if cells with mutations are able to divide further, and if the mutation favors neoplastic transformation, cancer can take place. Mutagenic effects on germ cells can lead to abortion or genetic disease.
Mutagenic does not necessarily mean an increase in cell proliferation always. All mutagens are not carcinogens, and all carcinogens do not act by altering DNA sequence!
Hope this makes it clear, that cellular toxicity and mutagenicity are major concerns when using chemicals for human use, but not so easy to understand without lot of background learning.
Genotoxicity is my area of interest hence can understand the question from your background!
The cytotoxic potency of mutagens is highly correlated with their mutagenic potency. A substance exhibiting low cytotoxicity could be at the most a weak mutagen. Mutagens are cytotoxic when they interfere with cell division. Mutagens can cause mutations, but whether mutagens can cause cell proliferation will depend on the ability of the cell to repair the DNA damage. This relationship implies that the maximum potential mutagenic potency of a substance may be reliably estimated from rapid and straightforward measurements of their cytotoxic potential.
The substance may exert cytotoxic effects by pathways independent of its mutagenic action. Differences in repair capability or fidelity may influence both cytotoxicity and mutagenicity. Cytotoxicity may result from genetic (DNA related) and non-genetic mechanism of injury. Substances that induce the highest mutation frequencies may have the tightest coupling between genetic injury and lethality which means cytotoxicity related to DNA damage predominate over other forms of general toxicity. Substances that induce lower frequency of mutations may produce a larger proportion of non-genetic injury.
Mutagenic NOT means cell increase. Mutations are caused by faulty DNA replication (especially meiosis) or faulty repair of DNA after damage (such as exposure to radiation or carcinogens)