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I feel ill or sick. Both mean the same. But when they modify a noun, do they mean the same? For example,

Sentence A: He is a sick person.

Sentence B: He is an ill person.

Do they mean the same?

I want to know the difference because 'ill' can mean 'bad' in some senses.

As you've already said, ill can mean bad in some senses. But so can sick. In fact, sick is the more common word if you want to describe somebody in the bad sense. (However, sick can also be used as a slang term for cool or awesome.)

Context determines the meaning:

That serial killer is one sick individual.

Ill intentions often result in ill deeds.

If you mean physically unwell, neither of your sentences would be normal. By adding person, you are implying the bad sense.

The best way to describe someone who is unwell, and have it be interpreted as you want, is to add the word feeling:

He is feeling sick.

He is feeling ill.

Or:

He isn't feeling well.

It's also common to say (in informal dialogue):

"He isn't feeling good."

In terms of describing being physically unwell, sick and ill are mostly used interchangeably.

But per Google Books NGram Viewer, sick seems to be the more common of the two words.