There is a very subtle difference between the two sentences. Hopefully by the end of this, you'll understand that difference at a gut level, even if you find it as hard to articulate as most native speakers do.
First, consider these sentences:
I feel happy.
I feel sad.
I feel blue.
I feel confident.
In these sentences, "feel" works as a linking verb—like is in "The door is open." They are all in the simple present tense, but they describe the current state of the speaker rather than stating a fact out of time.
Notice that they all have an adjective. The adjective describes the speaker. You could also say that the speaker is happy, is sad, is blue, etc.
"I feel independent" doesn't quite follow that pattern. Consider these sentences:
I am a slave at the Oak Alley Plantation, but I feel free.
"Doctor, I feel sick."
"Well, I've checked you out thoroughly, and you are healthy."
"I feel ridiculous wearing this expensive suit."
"Well, you look marvelous."
"When Gladstone tries to educate me about some matter of economics, I feel two inches high. But when Disraeli asks for my support in his latest parliamentary moves, I feel ten feet tall!
In these sentences, "feel" describes how the speaker seems to him- or herself. The speaker could be mistaken. They're a bit like this kind of sentence: "I feel that the new employee is not going to work out", but the adjective names a feeling rather than a proposition. The feeling could be wrong because it implies or suggests something about the speaker's outward situation. The speaker's feeling could be out of sync with how he or she really is.
Independence is the sort of thing you can feel but also be wrong about. So, "I feel more independent in my new job" doesn't mean that you are more independent in your new job. Maybe your previous job was freelance work, but you felt dependent on your clients. Regardless of whether you are or are not more independent at your new job, you feel more independent.
Now let's look at similar sentences with a noun instead of an adjective:
I feel something furry behind the sofa. [Said while feeling around with your hand behind the sofa.]
I think you can't know joy if you don't know sorrow. [Source.]
A moment or two later I too felt the sting of a bullet, and fell benumbed with pain. [Source.]
I am pure Ukrainian. My parents are Ukrainians. We don't feel independence yet, just a deterioration of living standards. But that's not connected with the concept of independence. [Source.]
Certainly in these sentences, "feel" doesn't work as a linking verb. The speaker isn't something furry, joy, sorrow, the sting of a bullet, etc.
These sentences describe feeling something as a way of experiencing something. Feeling by touch is the most fundamental kind of feeling, and the most direct kind of knowledge: direct experience. As used in these sentences, "feel" means some kind of direct interaction with something real and experiencing it that way. The last sentence is the (translated) words of a Ukrainian in 1992, the year after Ukraine's declared independence from the Soviet Union. The speaker means that even though Ukraine is independent, the people don't feel it yet: they don't feel the (real) independence of their country.
So, "I feel much more independence" suggests just a little more strongly that the speaker's independence is real than "I feel much more independent." It vaguely suggests that the speaker is actually doing more independent things at the new job, not just having emotions related to feeling independent. "I feel much more independent" emphasizes the emotion more, without regard for whether the speaker really is more independent.
If this answer has done its job, then you should feel like you sort of understand but you couldn't make a rule out of any of this. None of the above should be taken as a rule. People understand "I feel ____" by analogy with sentences like those shown above (or others). Which kind of sentence provides the strongest analogy depends on the context and on what goes in ____. Sometimes more than one of these meanings is a pretty good fit; the distinctions are not always clear.