The General Instruction of the Roman Missal, paragraph 382 notes:
At Funeral Masses there should usually be a short Homily, but to the exclusion of a funeral
eulogy of any kind.
Why is this? Is the Church trying to again be difficult? Nope. Everything the Church does is for the best of all people involved.
A lot of times, people want to give a eulogy at a funeral. Here's the problems that I've noted in my experience
1) The person giving the eulogy (and, by extension, any of their family who know they are going to be giving the eulogy) are nervous throughout the funeral and can't really simply enter into the prayers, readings, etc. because they are thinking about what they are going to say.
2) The funeral has a beautiful flow to it, and each moment leads to the next, and their is a wisdom behind the flow of a funeral Mass that comes to us from centuries and millennia of experience. EVERY TIME I've interrupted the flow to have everyone sit down to listen to a eulogy, it has felt like a healthy process and movement of the liturgy has been completely interrupted and thrown out of whack.
3) The homily (hopefully) helps the people get a good dose of authentic Catholic teaching and comfort about death and life. There is no way to control what is said in a eulogy. In addition to the potential of inappropriate things being said, there is also often times a spewing forth of a lot of false theology, song or movie quotes, confusing and frankly sometimes completely wrong understandings of death.
Because of all of this the place for a eulogy is at the showing.
Every family that I have helped guide toward doing the eulogy or eulogies at the showing instead of during the Mass has been thankful for the suggestion when it was all over with.