What is the difference between permutations and combinations?

The main difference is that order of objects matters for permutations. Both involve choosing k objects from a group of size n. Permutations are used to calculate the amount of possible ways a group can be made when a particular order or arrangement of the chosen group is asked for. Combinations are used to calculate the amount of ways a group can be made from a larger group when a particular order is not specified. An example could be choosing a subcommittee of 3 people from a larger committee of 20 - this would be a combination as the order of people in the subcommittee is not mentioned but if this subcommittee is the president, treasurer and secretary then it is now a permutation as the order is now specified.

The two are related by the fact that each combination has k! times the amount of order specific rearrangements of the subgroup to turn it into a permutation for the group of size k, so the number of permutations is always larger than the number of combinations for a given group size.