Usually, you would not care about both of them simultaneously. Depending on the goal of your analysis (say, description vs. prediction), you would only care about one.
- For description, multicollinearity is just a fact to be mentioned, just one of the characteristics of the data.
- For prediction, omitted variable bias is largely irrelevant as you are not interested in model's coefficients per se, only in predictions.
You may care about both of them at once when attempting to do causal inference. I will argue that you should actually worry about the omitted variable bias but not multicollinearity. Omitted variable bias results from a faulty model (a cause in contrast to the characteristics of the underlying phenomenon). You can remedy it by changing the model. Meanwhile, impecrfect multicollinearity can very well arise in a well specified model as a characteristic of the underlying phenomenon. Given the well specified model and the data that you have, there is no sound escape from multicollinearity . In that sense you should just acknowledge it and the resulting uncertainty in your parameter estimates and inference.