I will preface this post by stating that I tutor only adult college students; those who tutor small children will have different experiences.
I find that most students, especially those new to tutoring, are adverse to the prospect of online tutoring. Considering that many are taking online classes, I should find that strange. However, in many cases, their online educational experience is so poor that they cannot imagine being tutored that way as well. Of my students who have tried online tutoring, however, none has ever asked me to tutor them in person afterward. (I have been tutoring, principally online, since December 2008.)
Students experience at least four advantages from online tutoring. First, my students enjoy the flexibility of being able to learn in a relaxed environment, such as their homes or dorms, without a stranger within their terrain. (For numerous reasons, I will not tutor in a student's home or in my home, so we need to find a public location.) Likewise, the student saves the hassle of traveling to the tutoring location as well as the time (and if they don't have a car, the cost). Also, a lot of my students ask for tutoring when the public library is closed. We end of studying at a fast food restaurant, generally to the glares of the staff members when we buy drinks and spend two or more hours working on homework assignments. Furthermore, between the web conferencing software which I use and the communications software, we can see each other and the work as if were seated side by side. Finally, and possibly most importantly, I have accumulated a vast library of resources in the fields of accounting and finance; I could load the resources onto a thumb drive for an in-person session, but given the varying capacities of my students' laptops, it is far easier for me to access the materials on my own computer.
Generally, the greatest obstacle to online tutoring is that I may need to see the problems in the student's textbook in order to work with them; however, with the proliferation of e-textbooks and the widespread use of scanners, few students have these issues. In fact, some colleges post the homework assignments on their or the textbook publisher's website, making it easy for the student and me to view the problem on the Internet. Sometimes we experience Internet or power issues; however, I do not charge the student when these things arise. Also, if we experience communication issues, I have a business phone line on which we can talk.
In summary, I would strongly suggest that students who resist online tutoring to reconsider their decisions.