On a foundational level, Christian and non-Christian counseling differ on their fundamental assumptions about where truth is derived, and what it means to be human.
With all of counseling, whether Christian, or other, there really isn’t one approach. With psychotherapy, there are many theories and treatment techniques all varying in purpose, efficacy, and evidence. To give you an idea, within contemporary psychology there is cognitive behavioral therapy, dialectical therapy, eye movement desensitization and reprocessing, existential therapy, psychodynamic therapy, humanistic therapy, gestalt therapy, behavioral therapy, interpersonal therapy, and the list goes on. Each of these psychotherapeutic techniques has differing presumptions about psychology, philosophy, and biology.
Likewise, “Christian counseling” encompasses wide spectrum of approaches. I could sit here for hours and unpack the various approaches of different Christian counselors. The approaches you get will really depend on the denominational affiliation, and bents of the faith communities the "counseling" is coming out of.
To simplify things, I will really address two major categories of conservative Christian approaches. Note that there are chrisitan approaches that fall outside of these two broad categories. One of these outlying approaches is the goofy charismatic/faith healing stuff, which is largely divorced from historic Christian faith and tradition.
The first category is nouthetic, or what is now oftentimes referred to as “Biblical” counseling. Organizations like the Christian Counseling and Education Foundation (CCEF), the Biblical Counseling Coalition (BCC), and the Association of Certified Biblical Counselors (ACBC, formerly NANC), hold to iterations of this approach. The biblical counseling folks got their start with Jay Adams, who was prominent through the 70s and 80s. Adams rejected psychotherapy and psychology, and pushed for Christians to develop an approach to counseling that upheld the sufficiency of scripture and sought to find answers to counseling problems in the bible. It was a novel idea, and operated off the premise that if the bible is sufficient for life and faith, then it too should be sufficient for life’s counseling problems. There is little evidence on the efficacy of biblical counseling, however some of the current practitioners of biblical counseling are probably practicing a modified version of cognitive behavioral therapy (even though they wouldn't admit it) and are not close to Jay Adam’s original model.
The second common category of Christian counseling is an integrated approach. In this approach to counseling, Christians hold that the bible is the ultimate arbiter on the issues it speaks to, but these counselors are open to using psychotherapeutic techniques as long as they are brought under the authority of scripture. What you get from this camp is a real mixed bag, and even within integrated Christian counselors you will find a pretty broad spectrum of approaches here.
I'm going to end here because I am hungry and want to make dinner. But, if you (or anyone else) has specific questions, I will do my best to answer. I was going to add citations to the above, however I decided that would be overkill for a forum post. If you want acadmeic reading/citaions for stuff let me know and I can send you that stuff (please just make the material requests specific).