I would say that an algorithm is a private case of model because it describes how something generally works --- just mathematically.
Formally, an algorithm is "a specific set of instructions for carrying out a procedure". Bread is not an algorithm, but a set of instructions on how to bake bread is an algorithm. A formula is not an algorithm, but a set of instructions on how to solve a formula (such as PEMDAS) is an algorithm. Pi is not an algorithm, but a set of instructions on how to calculate pi is an algorithm. None of these algorithms are models.
A model is a known system which has been claimed to behavior similarly to an unknown system. An algorithm can be developed which then becomes a model, or a model can be proposed which is then tested by an algorithm. Conway's Game of Life is an algorithm about how to change a grid of 0's and 1's; it became a model when someone claimed that the 0's and 1's behave similarly to organisms. The Combined Gas Law is a model; an experiment designed to compare the predictions of this law to actual gasses is an algorithm. There are very few model that have no corresponding algorithms because when someone creates a model they usually propose a algorithm that would allow testing of the model. In principle, they can exist though.
When Democritus first proposed his theory of atoms, he was proposing a model. However, an algorithm that would allow comparison of this model to the actual world did not exist at the time. Eventually such algorithms were developed, but for a while there was a model that did not correspond to any algorithms.