It all depends on your goal.
If you want to know how many people smoke and how many people die of lung cancer you can just count them, but if you want to know whether smoking increases the risk for lung cancer then you need statistical inference.
If you want to know high school students' educational attainments, you can just look at complete data, but if you want to know the effects of high school students' family backgrounds and mental abilities on their eventual educational attainments you need statistical inference.
If you want to know workers' earnings, you can just look at census data, but if you want to study the effects of educational attainment on earnings, you need statistical inference (you can find more examples in Morgan & Winship, Counterfactuals and Causal Inference: Methods and Principles for Social Research.)
Generally speaking, if you are only looking for summary statistics in order to communicate the largest amount of information as simply as possible, you can just count, sum, divide, plot etc.
But if you wish to predict what will happen, or to understand what causes what, then you need statistical inference: assumptions, paradigms, estimation, hypothesis testing, model validation, etc.