I think the distinction is somewhat of an illusion, although, perhaps, a necessary category.
For every religion that we may delineate as political or apolitical, we can find examples of its application in the opposite sense.
The very first thing to remember is that the distinction itself
is a modern idea, and one that (ostensibly) exists solely in Western nations. Take, for instance, medieval western societies, which had no separation between church and state. The church IS the state, and the state IS the church.
To declare that Islam is political is to misconstrue the idea that religion and politics are separate entities in the first place, especially in areas where Islam is endemic to native culture. This is the prime conflict when Islam is exported to areas where we have an (ostensible) separation between church and state, such as Europe and North America.
I keep using the word "ostensible," and that is simply to underline that our notion of separation between church and state is and always has been in tension with traditional methods of governing. It is an ideal, which has fluctuating applications. We claim to uphold this ideal, but in reality, we simply do not notice how much religion is incorporated into our politics because it is ubiquitous. It becomes a stark reality when outside forces attempt to do the same thing (such as shariah law), because we simply tend not to agree with the proposed laws.
Even the Church of Satan
, whose ruling body makes no political statements, is, it could be argued, political; it is political in the sense that many members feel passionate about their political positions, and some even seek to promote these ideas. What is significant is that each member could be fighting for drastically opposing ideas, and both are considered equally Satanic in nature, because it is the individual Satanist that is active in the world, not the administration of the CoS. The administration of the CoS is so firm on this idea of them taking a neutral apolitical stance, that talk of politics is forbidden on this board (hopefully I haven't stepped over this line, as I am talking about politics in general terms).
Other religions can and do mandate certain political positions (unlike the CoS) and this is where we see various levels of political activities.
Now, I could see even any "apolitical" religion just being used politically...or an individual being a more moderate follower of a "political" religion and ignoring the holy law parts of it.
Every religious person on the planet edits their respective religion in ways that suit them best.
But, is there any meaningful distinction you find when we look at what the religion ACTUALLY says? Or is that not a meaningful distinction either, when you look at the countless interpretations of the same text?
Textual analysis does provide a meaningful distinction, but usually only to scholars! The great majority of religionists never actually read their texts, and when they do, they carefully edit the passages they consider meaningful.
I would say that a political religion would tend to be more violent than an apolitical one, due to the need for a whole system of political force governing territory, which would then be expanded.
I have never, ever, heard of a religion that did not
exist as a political statement, even if that statement is: "Let's withdraw from the world because it has gone to shit. We'll wait for the salvation/apocalypse/comet/aliens/etc. in our secluded commune/church/mountaintop without the pollutant of the outside world."
The above is still making a political statement; it is judging the world as corrupt, and providing an alternative.
Religious ideas are born within and responding to specific historical circumstances, and as such they are always political.
Your question is more one of evaluating the level of active political movement within particular religions, and the answer to that is: it varies.
Even with Islam, the root is much more than a religious ideal (although that is usually the focus), but more of a conflict between internal struggles (the old ways falling our of favour, and the news ways gaining more and more popularity) and external pressure (outside forces threatening to take power from current leaders) that causes the most violence.
Historically, the trend for violence is almost always a combination of this dual factor of external and internal pressures, which increases tensions, which then erupts into violence, both externally and internally.