Dictionary definitions can only hint at what these words mean. I'll illustrate them with a few well-known examples, a brief look at their etymologies, and lots of links to real, typical uses of the words. Real usage in context will give you a better understanding of these words than any abstract description.
Subtle comes from Latin roots meaning "under the web" or "under the weave" (sub-tilis, related to "textile"), suggesting light, delicate fabric made of fine threads woven precisely. In English, this has given birth to a variety of extended meanings, some related to fine consistency, some to the keenness of mind needed to produce or perceive it, and some to both.
A "subtle distinction" is one that is very slight, so that you must pay very close attention to understand what makes the distinguished things different, such as the distinction in law between larceny and fraud. A "subtle odor" is one that can barely be perceived, because its concentration in the air is so low. A "subtle operation" is an action that requires precision and delicacy rather than force to be successful, such as assembling a watch or persuading a temperamental person to make a decision. Or a "subtle operation" can be one that works in ways that are hard to perceive or understand even if you're watching the action closely (it happens "under the weave"), such as the ways that power and status are established in social groups.
A "subtle mind" is one that perceives subtle things or devises subtle plans to achieve results in non-obvious ways. Subtlety of mind suggests perceiving or understanding deeply rather than superficially. It can also suggest a kind of untrustworthy cleverness—sneakiness. Perhaps the most famous use of the word "subtle" occurs in Genesis 3:1:
Now the serpent was more subtle than any beast of the field which the Lord God had made. And he said unto the woman, "Yea, hath God said, Ye shall not eat of every tree in the garden?"
The serpent is tricking Eve into eating the forbidden fruit, by subtle arguments and subtle manipulations. Other translations give a variety of words in place of "subtle": shrewd, clever, crafty, cunning, astute.
Discriminate comes from Latin roots meaning to make a difference or discern a difference between things. In its most primary sense in English, "discriminate" simply means to perceive a difference between things that are similar, enabling you to treat them differently. For example, this page talks about experiments with octopuses, in which they were found to be able to discriminate between mirror images, and could more easily discriminate between horizontally oriented figures than between diagonal ones. In mathematics and science, a discriminant is a factor that is relatively easy to calculate or measure, which enables you to sort similar things into separate groups, like the way in algebra the expression b2 – 4ac tells which kind of roots a polynomial has.
More commonly, though, people use the word "discriminate" to mean noticing relatively subtle differences between similar things, which make one thing better than another. For example:
Kushner decided to create a caffeine-free substitute that would taste so much like coffee, it would satisfy those with discriminating taste while still offering health benefits.
That is, some coffee drinkers are aware of and care about subtle differences in flavor: they have discriminating tastes. A discriminating coffee drinker would most likely be displeased with a decaffeinated coffee, because removing the caffeine also tends to alter the flavor. Calling someone discriminating in this sense is a compliment. Calling someone indiscriminate is pejorative: it suggests obliviousness to quality, subtlety, or refinement. A person who drinks coffee indiscriminately doesn't notice the fine differences in flavor between different varieties of coffee bean or methods of preparation.
In more recent times, at least in the United States, the word discriminate has taken on a new, narrower sense that now almost overpowers the others. This sense is: treating people differently in public matters such as hiring employees, serving customers, providing public transportation, or providing housing, on the basis of matters of group prejudice or group animus rather than on the basis of relevant matters such as qualification to do the job. When people speak of racial discrimination, this is what they have in mind. The Civil Rights Act of 1964, Title VI says:
No person in the United States shall, on the ground of race, color, or national origin, be excluded from participation in, be denied the benefits of, or be subjected to discrimination under any program or activity receiving Federal financial assistance.
Many other public and private organizations have adopted similar wording for their own anti-discrimination policies.
Now that you've seen how "subtle" and "discriminating" are used in practice, you can form your own conclusion about whether they are synonyms. In my opinion, it would be misleading to call them synonyms, though there is a small overlap between a couple of their senses. Both have senses suggesting refined tastes or refined perception. But hopefully now it doesn't matter, because now you've got your understanding of the words grounded in widely known real usage more than in definitions.