Some good points were mentioned already. A big issue with using DSLRs for video is that the lenses can be noisy, which is picked up by the on-board microphone. To overcome this, you would need to buy the special. Quiet lenses, such as Canon's STM (STepper Motor) models are available, but that means skipping the less-expensive kit lenses and shelling out more money than you might have planned. For example, the Canon EF-S 18-135mm 3.5-5.6 STM is a $550 lens at Amazon.
If your needs are purely for home video purposes and the budget allows, I'd still go with a DSLR with a quiet lens. You'll get the benefit of being able to take great stills when desired as well.
If you want to dabble in making advanced amateur movies or video for hire (weddings, etc.) you will definitely need a true video camera with real capabilities, such as XLR input, follow-focus, etc.
Finally, be sure to check the recording limitations of the camera you are considering. Due to sensor overheating, many models limit recording to no more tha. 30 minutes at a time, even though your memory card might have plenty of capacity for longer recordings.