A conclusion comes at the end of a thought process. It can involve developing a summary of what you've gone through so far to reach that conclusion and expressing a decision going forward. In negotiations, the conclusions are what you come to in reaching a final settlement. In science, conclusions are your interpretations of the data in an experiment.
An inference requires moving from information of some kind to a generalization. For example, if you come in the house and see your living room furniture torn up, you might infer that your new puppy got out of the crate and shredded everything. These interpretations do not represent your final opinion on a matter; they just help you get there.
In the case of the torn-up living room, there are other pieces of information that you might need to reach a final conclusion. For example, if you still near the puppy barking in the laundry room, where the crate is, your first inference might be incorrect. The broken window and the rifled filing cabinet might mean that someone was searching for something in your house -- so your first inference may be incorrect.
It can take several inferences before you can arrive at a meaningful conclusion. Inferences represent a step in the process from collecting data or information to rendering a final judgment on that information, and to making a decision about how to respond.