Reiki healing techniques and reiki energy have become fairly mainstream these days. Most people have an understanding of what reiki healing is, or at the least have an understanding that it's a form of energy work. Not many people know what trance healing is, much less how the energy used differs from reiki energy.
Reiki healing is a passive form of energy work. When the practitioner sets the intention to work, reiki energy flows from a divine source through the practitioner, into the client. The energy flow is not directed by the client or practitioner. It simply flows through the body, clearing and unblocking anything in its way, re-balancing what may be out of balance (particularly in regards to the chakras).
Trance healing is actually a form of trance mediumship. The practitioner has formed a relationship with one spirit guide, who has the practitioner's permission to enter his/her physical space. The practitioner goes into a trance, allowing the guide to enter their auric field and part of their physical body. In traditional mediumship, the guide would then speak through the practitioner. In trance healing, however, the guide lays a hand on the body of the client and directs healing energy into the client. Trance is a more aggressive form of healing. The flow of energy is directed consciously by the guide.
In my experience with my own practice, I have found reiki is ideal for those who are having trouble releasing emotional blocks, feeling out of sorts, or just going through a difficult time and want to keep an even keel during the hurdles of their lives. Trance healing seems to be best suited for chronic or terminal illnesses, or people who are at the end stages of life. I have found animals are usually resistant to trance healing unless they are at the end of life as well. Animals can tell the energy they feel isn't coming from a human being, and this usually frightens them. Reiki is often best for animals under most circumstances.
I do want to clarify, that this has been my own personal experience with my clients, and shouldn't be taken as gospel. Healing of any kind is a highly individual process. It's something that begins with a client's desire to heal. If you feel drawn to a certain healing technique, either to receive it or learn how to use it, by all means explore!