In AI (but in general too, I believe), a simplification is that modeling is more akin to Mathematics (and related hard sciences involved, like Physics and... Computer Science), and implementation to Software Engineering.
Let's take a concrete example, really outside of AI: Find the minimum value of a given polynomial, if it exists.
The Mathematician will derivate the polynomial, find the zeros, and checkout convexity to find a minimum (if there is any zero). This procedure is very standard---some will say straightforward. It relies on a body of knowledge and an abstraction level that is appropriate for manual proof.
The Software Engineer approach is actually way longer to explain, and I am going to skip it. The point is that the body of knowledge is related but different: We have to find now a step-by-step procedure for the computer to achieve the result. The Mathematician one could be implemented directly in MathLab, almost verbatim, but we assume MathLab. And to build MathLab, we are back to the problem of making a procedure the computer can execute. We could for example base a procedure on Euler's method to find roots (a "simple" approach that closes on roots step after step), etc.
Simple mathematical operations can be quite complex to implement on a computer. Perhaps the most famous is random-number generation. Mathematically, the concept is pure and clear. Generating an actual random-number is more elusive than it looks, to the point it calls for new models and new implementations...
A concrete example from history: Neural networks. In the 80s and 90s, NNs were weighted graphs that could be executed on computers using graph libraries or similar foundation libs. Choosing the weights was challenging. One day the back-propagation learning model was introduced to automated the choice of weights. The model relied on a procedure dedicated to NNs, using a terminology like partial derivates, gradient descent, chain rules, etc. And later then, clever engineers created libraries to automate the back-propagation procedure. The libraries can be somewhat far from the original model, as engineers learn how to make it computable, even faster (i.e. optimization, approximations/truncations).