As per comments, there's no real difference between the two idioms.
In other contexts (such as programming) an instance of something means an incarnation, actualisation, realisation (a real thing, as opposed to a definition of that thing). And an example can often mean an instructional case (perhaps specifically created for educational purposes).
But both for instance and for example are used in exactly the same way to introduce a specific illustrative case (or cases) - often with the implication it's just a random selection from many possible alternatives.
OP has already established that for example is more common (as my link shows, it's been increasingly displacing for instance over many decades). Initially, I was prepared to believe this might justify saying that for instance is "more formal".
But in fact I don't sense that myself, and I find no evidence for the idea. Consider these usage figures...
1a: for example I have (1,100,000 written instances in Google Books)
1b: for example I've 104,000
2a for instance I have 704,000
2b: for instance I've 76,500
In both cases the "informal" contracted versions account for about 1 in 10 usages, which strongly suggests no significant difference in terms of formal/informal register.
So my advice to OP is to accept for instance as an exact synonym whenever he comes across it, but to stick with majority/modern usage and always use for example in his own speech and writing.
To use X - X here has to be a tool, and to use X will mean do an activity where X will help or is needed to accomplish that activity. E.g.: Use the pen to write down what I say. A pen is a tool that is needed to accomplish writing.
To make use of X - This is like to use X except that X is not necessarily needed to complete the activity in question, but it will help. E.g.: Feel free to make use of anything in my spice rack when you start cooking. The spices might help, but aren't strictly required.
To tap into X - A physical tap is something that allows one to draw out or extract water, liquid, or other similar resource from a stream, network, or container. Following this logic, tapping into something can mean attempting to draw out or extract something needed (from someone) that can help you complete an activity, and this something will not be a physical object but a mental resource such as knowledge, skill, social connections, etc. Tap into your customer service skills when dealing with difficult requests.
So ...
A market in my opinion at least, that can be tapped into. Graduates would more likely be something made use of unless your location is starved for the skills of this specific group of graduates. Tap into X somewhat carries the subtext of X is needed by us but not vice versa.
And,
Can you make use of these papers?
Can you use these papers?
The difference in meaning depends on what the papers contain and how useful they are to the person you are offering to them. If you think they will help, but aren't required, make use of is appropriate. If you think they are needed by him to accomplish his task, use is appropriate.
Best Answer
These are mostly interchangeable - but according to my gut feeling, "for instance" is for describing a hypothetical situation, and "for example" is for giving examples. You can use either without being too off. For instance, you could give an example of something material, and then you would say, "I need some English flash cards in a nice, durable material, for example, laminated cardboard".