Is EMDR the same thing as hypnotherapy?

My understanding is that they share the same roots! EMDR and NLP both grew out of Hypnotherapy. Both EMDR and Hypnotherapy work by training the client to access internal resources and ability to calm their nervous systems. With this foundation, painful, uncomfortable and traumatic memories can be processed without going into a fight-or-flight response. EMDR has the additional benefit of bilateral stimulation, which aids the processing.

No, they are not same. EMDR is a technique that uses alternate-hemispheric stimulation of the brain to cause a disruption of the automatic circuit that is being triggered and fired as a result of past trauma. The disruption of the circuit in the brain causes the neurons to "unwire" so that the pathway no longer fires when you are exposed to the stimulus. In other words, you don't react automatically to situations that used to trigger you. You can remain calm and composed.
Hypnotherapy is a process that has a client go into a trance. The therapist then makes suggestions and talks directly to the subconscious of the client to effect changes in their behavior.
The two techniques together make for powerful transformations for clients.