Consider following example: your IV is gender and the DV is smoker/nonsmoker. The two variables would be considered "statistically" independent if you wouldn't find a relationship between gender and smoking status, meaning that you would have identical or similar percentage distributions. On the other hand if you found that men are more likely to be smokers than women, then what it means is that whether one smokes or not will depend to a certain degree on whether you're aman or a woman (men = 70%smokers, 30%nonsmokers: women = 30%smokers, 70%nonsmokers), the two variables are dependent, you could say there is relationship between the two. Going back to what you wrote... statistically dependent means that the distributions are not identical and not the other way around.
We do reject the nullhypothesis (there is no dependency (independence) between IV and DV) in favor of the alternative hypothesis (there is a dependency or relationship between IV and DV) when we find a statistical result in favor for the alternative hypothesis. The nullhypothesis is not necessarily rejected when proportions of the cell are nonidentical. Two factors contribute to the rejection of it, one being the sample size, the other is the relative discrepancy among the cells. The larger the discrepancy the more likely you will reject the nullhypothis, increase the sample size and with the same discrepancy you will be even more likely to reject it.
Hope that helps.