When I consider what the phrase "spiritual experience" means, I end up answering that question in a bit of a round-about way. Please bear with me while I get there.
I realize that I can construct a reasonable, intellectual argument for or against anything. I mean that fully. If I decide to construct an argument that casts the Church negatively, I can do so; if I want to cast the Church positively, no matter the issue, I can do so. Therefore, I have made a conscious decision to look actively and passionately for a way to reconcile difficulties and remain faithful.
In college, as part of my research on Manifest Destiny, I read just about every anti-Mormon writing of the 19th Century. I took some classes at the Harvard Divinity School - not exactly a bastion of extreme conservatism or champion of Mormonism. After graduation, I lived in the Deep South for a few years. I am probably as well-versed in anti-Mormon rhetoric as most, so my statement in the last paragraph is not stated carelessly. I also, however, understand that I can learn MUCH about the Gospel of Jesus Christ even from classic anti-Mormon preachers and denominations. A few of the most profound spiritual insights I have received have come by hearing something I already believed phrased differently by someone who believes I am headed straight for Hell and would dance in the streets if Mormonism was eliminated completely - simply by hearing it from a different perspective I had never considered previously.
That, in my mind, is the key - truly internalizing and trying to live the Articles of Faith and the core principles of the Gospel, especially by developing the characteristics of godliness outlined as the pathway to perfection in the Sermon on the Mount. I have had more truly spiritual experiences in the last few years, as I have intentionally and purposefully pursued that objective, than I had in the previous twenty years of my life - including in the various leadership callings I have had in the Church.
So, what is a spiritual experience?
It is an experience that makes my spirit grow - that brings me closer to the Father and the Son by making me more like them.