A couple things follow from this. First, where legitimate children were under the patria potestas of their father, illegitimate children had no famila and were in no one’s power unless their father or some other man legally adopted them. Second, they inherited their status and citizenship from their mother. So if she was free, her children were freeborn. If she was a slave, so were they. If the mother was a Roman citizen, so too were her children. So, for example, if a free citizen woman had children with a non-citizen man,** they would be free citizens, regardless of his free or slave status.*** Third, since they were legally their mother’s offspring but not their father’s, under the Leges Iuliae they counted towards the number of children she needed to be free from guardianship, but did not contribute to the benefits a man could get from having more children.
Since they weren’t part of their father’s family, illegitimate children couldn’t inherit from him if he died without making a will. And until the second century CE, the same went for the inheritance of a woman’s property (except her dowry) by any of her children, legitimate or illegitimate. Unless she made a will, any of a woman’s children had as much right to her property as her parents and siblings, sometimes less. It was only in the second century CE that the law changed, giving children priority in the inheritance of their mother’s property if she died intestate.
*A married woman’s husband had the right to choose whether to raise a child or expose it. Children he raised were usually considered to be his own.
**Since under Roman law marriage was only between two citizens, the children of any relationship between a citizen and a non-citizen would always be illegitimate.
***Marriage-like relationships between free women and slave men were common enough to seriously concern certain moralists, especially 2nd-4th centuries. These de-facto marriages, along with those between two slaves, were known as contubernium.
Gardner, Jane F. "Legal Stumbling-Blocks for Lower-Class Families in Rome." In The Roman Family in Italy: Status, Sentiment, Space, edited by Beryl Rawson and Paul Weaver, 35-54. Oxford: Oxford University Press, 1999.
Rawson, Beryl. "Adult-Child Relationships in Roman Society." In Marriage, Divorce and Children in Ancient Rome, edited by Beryl Rawson, 7-30. Oxford: Oxford University Press, 1991.
Pomeroy, Sarah B. Goddesses, Whores, Wives, and Slaves. New York: Schocken, 1995.