The question in the title to this post is the number one most asked question I hear. The real answer to this question keeps evolving over time. Computer programs change over time and some disappear. Online websites change almost from day to day. So what was true and which program was the "best" have been changing regularly. Here are some of my responses to the question.
Before outlining the issues I need to point out that from my perspective as a member of The Church of Jesus Christ of Latter-day Saints, I will sooner or later put all of my genealogical research on the FamilySearch.org Family Tree for reasons that involve my personal beliefs.
1. If you are using an old version of a program or a discontinued program such as Personal Ancestral File, you are asking for trouble and data loss. It may not happen, but using an old computer program is not like driving an old car. It could break down anytime and anywhere. The adage about wear it out and use it up does not apply to computers. The principle here is called "Data Migration." This means that you keep updating your files so they work with newer computers and newer programs. For example, I just updated by Apple OS X system to El Capitan Version 10.11.4 as soon as it became available. The longer you wait to upgrade, the more painful the experience may be for your existing programs and data files.
2. Members of the Church presently have free access to four large, online database programs, each of which support an online family tree option. These are FamilySearch.org, Ancestry.com, MyHeritage.com and Findmypast.com. Any one of these programs would provide a perfectly adequate place to store your family history information. The FamilySearch.org Family Tree is often criticized because it is subject to change. This criticism is short-sighted and ignores the basic nature of the program, but those concerned about changes could use any one of the other three programs that provide for personally maintained family trees.
3. There are any number of very good local, genealogical database programs. If you want a list go to the GenSoftReviews.com for a long list of programs of all types and on all platforms. Those who advocate having your own personal program have a variety of reasons that include maintaining private information, information about living people and other related issues. Most of the reasons for maintaining a personal program as opposed to using the FamilySearch.org Family Tree are equally applicable to the online programs. So this concern revolves around the confidence that the user has in putting their information online as opposed to their confidence in their own computers and programs.
4. I see the issue as one involving time constraints. I will not live forever and I have more research than time. I cannot afford the time it would take to duplicate my work between the FamilySearch.org Family Tree and any other program. Personally, I am moving all of my information to the Family Tree.
5. The ability to move information between the various genealogy programs is presently very limited. There are a handful of programs that connect to FamilySearch.org's Family Tree, but there are only very limited ways to move information to and fro between any of the other programs. This fact is the largest limiting factor in working with multiple program whether online or on one computer.
6. Some people are constrained by personal fears concerning ID theft and loss of privacy and will not put any of their genealogy online. I have written at length about this issue in the past and my opinion of this issue has not changed at all. I believe that the privacy and ID theft concerns arise from an ignorance of the amount of information available about anyone in the Unites States and a media created fear driven primarily by those who exaggerate the risks involved to promote their own interests. Putting information about dead people online does not involve either concern any more than writing a biography or history book.
I maintain more than one separate copies of my ancestral information. I keep only one copy, the FamilySearch.org Family Tree, completely up-to-date. I update the other programs from sources as they are discovered. I then transfer as much information as I have time to spend into the Family Tree.
I fully recognize that there are those who will take different positions based on their own experience and goals.