As a licensed massage therapist I'll tell you there is no difference. Its all marketing. Sports, orthopedic massage and medical massage are very similar. Taking a course in any of these will cover the majority of what is taught in the others. Deep Tissue is also a misnomer in the sense that people expect it means deep pressure. This is not true. Deep tissue simply means affecting deeper layers of tissue but you can affect deeper layers of tissue with Swedish. Also myfofacial release is a deep tissue modality but does not feel deep. If I had to give definition of each I would say;
Deep tissue is a therapeutic type of massage that utilizes many different modalities and techniques to address specific concerns. Deep Tissue can also be used as an umbrella term that includes all modalities that focus on specific work to alleviate specific symptoms. In this sense deep tissue can be as light as myofascial release or as deep as trigger point.
Sports massage is any massage performed on athletes (professional or armature). It can encompass techniques from a wide range of modalities such as Swedish, trigger point, lymphatic drainage, myofascial release, Thai and so on. What makes sports massage sports massage is the client base, not the applications used.
Most of these terms are useless and meaningless to you as a client. I can get my license to provide CEUs teach someone a few stretches and hand them a certificate that says they are a sports massage therapist. Legally there is no requirement to have any type of training in any specific modality (breast massage being the exception in Texas). However if I were to use a technique I was not trained in and on someone who was contraindicated for it, and that person disclosed that contraindication to me, I can be held liable for malpractice. If you are looking for a skilled therapist look for someone who has a good understanding of anatomy, and let them know your specific complaint. Ask as many questions as you feel you need before you are comfortable hiring that person. In my experience the best therapist are anatomy junkies. We are fascinated by the body and always want to know more. The ones who couldn't care less about anatomy should stick to Swedish, which is very valuable in its own right.
A good example would be of a therapist who ask questions like:
When did the pain start?
Describe the pain, dull, shooting...
Is it persistent or does it start and stop?
Where does the pain begin and where does it end?
Are you feeling pain in this body part or that?
How is your range of motion in (blank) joint?
What do you do for a living?
What do you do in your leisure time?
What other professionals have you seen before? What did they say?
What are you past injuries?