My understanding is as follows:
Filho (a) natural means that the parents, who are not married to each other, nevertheless have no impediments to marriage, i.e. neither one is married and they are therefore free to marry each other if they so desire.
In the 16th and 17th centuries one often sees couples getting married who already have a child (or more) who have been waiting for dispensations from the prohibitions relative to consanguinity, before they would be allowed to marry.
The term filho(a) natural can only be properly applied if the identity of both parents is known. (How else would one know that the parents were free to marry?)
Filho (a) illegítimo (a) means that at least one parent is not free to marry the other parent, e.g. one of them is married to somebody else.
Pai incógnito (and in some rare cases mãe incógnita), pai não sabido, means the identity of the father (in some rare cases the mother) is not known so that no judgement can be made as to
the status (ilegítimo vs. natural) of the child.
I could be wrong, but this is my understanding. I hope I have not confused anybody.
Illegitimacy (in Massachusetts where I live) was a term applied to any child born to a single parent. Therefore, we
sometimes misapply the term to Portuguese children whose parents, though not married, have no impediments to being married. (filho(a) natural and filho(a) ilegítimo(a) are not synonymous) Today, in Massachusetts, the term illegitimate, no longer appears on a contemporary birth record, though the marital status of the mother (and father, if known) is stated. If the identity of the father is unknown, the space is left blank. Those records are restricted and are only available to the legal guardian of the child (usually the mother, but not always, e.g grandparents, etc., or the State), or to the subject of the registration if he/she is an adult
John Miranda Raposo