The short answer
For most funerals, eulogy and funeral speech mean the same thing. They are used interchangeably by funeral directors, families, and clergy.
If anyone has told you the two are different, they are usually distinguishing one of two things.
A eulogy is sometimes used to mean specifically the speech delivered about the person who died. It comes from the Greek word for "good words" and is, at its heart, a portrait of the person.
A funeral speech can sometimes refer more broadly to anything spoken at the funeral, including welcomes from clergy, readings, prayers, the closing remarks, or short tributes from many speakers.
In practice, when a family asks you to give a "funeral speech" or a "eulogy," they mean the same thing. They are asking you to stand up and say something true and dignified about the person who died.
The slightly longer answer
Some traditions and some venues do draw a distinction.
In some Christian traditions, especially Catholic and Orthodox, the funeral mass has a specific structure that does not include a eulogy at all. The homily is given by the priest. A short tribute may be permitted at the end, or at the wake the night before, or at the graveside. If a priest tells you "we do not have eulogies at this mass," that is what they mean.
In some Jewish traditions, the eulogy is called a hesped and is delivered by the rabbi or by family members at the funeral itself.
In humanist or non-religious services, the eulogy is usually the centrepiece of the service.
If you are unsure what is appropriate at the specific funeral you are speaking at, ask the funeral director or the celebrant. They will tell you what the structure of the service is, what is expected of you, and how long you should speak for.
What to actually write
Whichever one you are giving, the writing is the same.
Pick three specific things you want the room to remember about the person. Build the piece around those three things.
Write one sentence that captures who they were. Not a list of accomplishments. A description of the person. "She was the kind of grandmother who made every grandchild feel like the favourite."
Tell two or three small specific stories that show that one sentence is true. Concrete scenes. The kitchen. The chair. The phone call. The thing she always said.
Close with what they leave behind. Plain words. Not sympathy card phrases.
That structure works for a eulogy in any tradition, and for a funeral speech of any length, in any venue, for any kind of service.
How long should it be
Three to five minutes is the standard length. About four hundred to seven hundred words.
Some traditions specifically ask for shorter, two or three minutes. If the funeral director or celebrant gives you a time limit, take it seriously. The structure of a service depends on every speaker keeping to their time.
Some humanist or non-religious services give one speaker more space. Five to seven minutes can be appropriate. Anything beyond seven minutes starts to lose the room, even from a son or daughter.
What about a tribute, a few words, or an obituary
These three terms are sometimes confused with eulogy and funeral speech.
A tribute is usually a short spoken acknowledgement, often two minutes or less, sometimes given by several people in turn at a service.
A few words is what people often say when they are inviting you to speak briefly. It usually means one or two minutes.
An obituary is a written notice published in a newspaper or online, usually before the funeral. It lists facts: name, date of birth and death, family, key life details, service information. It is not the same thing as a eulogy.
The eulogy is the spoken portrait. The obituary is the written record. They serve different purposes.