Generally someone who became deaf after learning to speak is called post-lingually deaf. One can’t say anything absolutely but if they lost their hearing after living as hearing person for a number of years, especially lost their hearing as an adult then they are a hearing person who can’t hear. Their identity was formed as a hearing person. Not many will learn sign language because they have lived most of their lives in the hearing world and all their friends and family are still in the hearing world. It can be very difficult.
Someone who is born deaf or becomes deaf at a very young age is more likely to identify as a deaf person, and depending on the social circumstances identify as a Deaf person with a central attachment to the Deaf community, using sign language. On the other hand they may be raised in the “oral” tradition where they do not learn sign language and instead their training is focused on speech reading (“lipreading”) and speaking. Some people who are raised in the oral tradition are very successful at developing speech that others understand
and/or at being able to understand speech by so called “lip reading”. Others never are able to develop one or both of those skills.
The groups they would belong to generally are different. The deafened adult might belong to
ALDA. Someone who has grown up identifying in the Deaf world would maybe belong to NAD, be active in local Deaf clubs, and Deaf sports leagues etc.
This can vary a great deal, especially depending on whether someone who is deaf grew up around other deaf people, for example attending a school for the deaf or has deaf family members (most do not have deaf family members), or instead grew up rarely if ever seeing another deaf person.
Another emerging group are Deaf people who have had cochlear implants which turned out to be successful in allowing them to decode some speech auditorally. Some implants end up with a successful outcome and others do not, depending on a number of factors. These are Deaf people, having had their identity formed as a deaf person, who can hear. Are you confused yet?
The one universal I would say is that all of them experience discrimination and as with other populations end up with someone outside their group trying to make decisions for them or making judgments about them. (Like Protestants explaining what Catholics believe, or Catholics explaining what Orthodox believe

)