If epistemology is the philosophical study of knowledge and ways of knowing, and if science is the ultimate system for the acquisition of knowledge, my question is simple: What is the difference between the two and why aren't they the same thing?
Science is NOT "the ultimate system for the acquisition of knowledge". Probably you are talking about the scientific method, but the scientific method is not what science is.
- Science is a type of knowledge , and
- Epistemology is the discipline that studies knowledge.
Science deals with a specific type of truth: empirical truth. That is far from being the absolute truth, which is more or less the target of philosophy. Empirical truth is anyway essential to our survival, a lot more than absolute truth. As long as we imagine that the world is made of small balls, and by applying the Boltzmann equation to those balls, we can predict a behavior, that's enough for us; that helps us surviving. Even if we came to know that the world is not made of small balls.
Creating knowledge implies an interaction between a subject (the observer) and an object (the observed). Ontology tries to focus the object, excluding the subject, but traditionally, such approach creates a problem: much of our understanding is biased by our capabilities (senses, reason, perception, etc., see Kant et.al.). That means that ontology is an attempt to get an absolute knowledge of the object, excluding the subject, which ends to be fallacious. For what I know, "ontic" is the extreme absolutism on the object, that would imply that apples would be red even if humans would have never existed.
Probably that's the essence of epistemology. Knowing an object does not imply only observing the object, but also the subject. Such approach is known as epistemologic. Normally, the deeper the analysis is, the larger focus on the subject will be (some suggest (e.g. George Berkeley) that objects don't exist, all are just productions of reason). So, the object is left aside, and the subject might be focused entirely. Such approach is called epistemic (simple words explanation, you can dig way deeper just by googling).
As you can conclude, epistemology is probably the frontier where science ends and philosophy begins. The approach of epistemology cannot be scientific, ergo it would focus a deeper truth, not just an empirical truth.
Why aren't they the same thing? Two white wines can taste the same for an uneducated palate. Do they taste the same?
 Which type? the one that is obtained using the scientific method, that is objective, etc.
If epistemology is the philosophical study of knowledge and ways of knowing, and if science is the ultimate system for the acquisition of knowledge
First off, not all knowledge is scientific. When I learn about how to read and write, or establish I am late to class, or even study philosophy, I am not studying, doing etc. science.
So epistemology may not be only the study of science.
Next up, science is not just knowledge. Science - the dictionary ably informs us - is an intellectual and practical activity. Scientists sort chemicals, do tests, and devise theories, and not all this is a way of knowing, knowledge.
So "science" cannot just be epistemology.
The question of what science is, is often called the "demarcation problem". Epistemology is defined by dialectic, by which I mean the history of philosophy.
When collapsing all ways of knowing into science, philosopher are oft accused of "scientism". Quine's 'naturalized epistemology' has been labelled this (dunno where the term scientism comes from / who coined it)
Trying to list ways those are different would be senseless, I think: How are apples and oranges different? They are both fruits, but...
“Science“ is both a methodology and a body of current work using that methodology. Epistemology specifically concerns the evidence-gathering aspects of that package, while Metaphysics is about principles of logical model building that such evidence helps us construct. Philosophy of Science includes both forms of methodological consideration, specific to the context of scientific practice.