There is an attempt being made at present to distinguish between the two terms in the title of this post.
Without clarifying the difference it is simply asserted that Shia-Sunni fighting is one thing one thing and that the "insurgency" is another. This is an assertion of a distinction without a difference.
The "insurgencies" in Iraq are variously:
-international jihadis who are all Sunni Muslims.
-local Iraqi Islamists (all Sunni Arabs)
-Tribals (Sunni Arabs)
-De-mobilized soldiers (overwhelmingly Sunni Arabs)
-Nationalists and Baathists (some Sunni Arabs and some secular Shia)
What are all these groups fighting for? With the exception of the "Nationalists and Baathists" who want to see the restoration of a secular and unitary state, the rest of them want to see the maintenance of the previous (a thousand years) disproportionate political power of Sunni Arab Muslims in Iraq. The international jihadis would express this as a desire for Sunni Muslim control without a focus on the Arab part of the identity.
What is different in the present "sectarian violence?" The Shia have now reciprocated in doing violence to their sectarian enemies.
That is the difference.