DVR and TiVo are both used in order to record the shows that are being shown on television so that you can watch them even after the show is over. There are still some debates about the differences between the two. DVR is said to be more of a remake of TiVo probably because TiVo came out first. Another difference between the two is their hard drives.
TiVo will just have an advantage because what you have recorded can be easily transferred to your chosen device. For example, if you have a MAC or a laptop, you can easily watch your videos there easily. For DVR, you still need to connect it before you can watch what you have recorded.