Not really sure it is enough for an answer, but feels too long for a comment; as with here, it primarily seems to be people who feel that the label "Christian", regardless of it's origin and literal meaning, has too many associations (perhaps more in the people they interact with than themselves).
As an example, there are phrases often used in media and politics such as "Christian values", "the Christian thing to do", "the Christian view", etc. As is demonstrated daily on this site, there is no such thing as a single definition of "Christian values", or "the Christian thing to do" - it is a huuuuuuuuuge spectrum.
Hence my interpretation of this is that these people are believers who are very happy to follow the teachings of Christ, but for whom the term "Christian" has become loaded; while it represents positive things, the same term is also used on their behalf, and without their consent to (mis?)represent them, or to form assumptions about their view on things simply from the label "Christian", when simply asking them could be more helpful to all parties.
An interesting feature of this, for example, is in politics - where "Lobbyist X" who happens to be Christian, cites some Christian head-count number, and uses that argument to support their case. However, it is not true to say that just because someone is Christian means that they agree with the view being presented by the lobbyist - it is false representation. Without trying to walk head-first into another label, lobbyists using such tactics tend to be trying to push through Christian-right/conservative perspectives. As a current example, some Christian-right folks are currently getting very worked up about the thought of proposed US laws relating to contraception and employer obligations - somewhat ignoring the fact that the vast majority of Christian women have used contraception (this is meant as an example only, so let's not get overly distracted on that).
Labels aside, I doubt you (as someone who is happy to identify as "Christian") would have any important belief differences. In terms of opinion differences - that is entirely the point: you'd need to ask them, as forming an opinion (of either party) merely from the label "Christian" can be very very misleading. As such, though, it seems a little bit implicit that a "follower of Jesus" / "follower of Christ" is probably not overly conservative in their views. The irony here is that avoiding one label ("Christian") seems to imply another ("liberal") - but that might just be me.