I think the following explains it pretty good.

What Is a “Rhema”?

Scripture for personal application

Logos—The Word of God

There are two primary Greek words that describe Scripture which are translated word in the New Testament. The first, logos, refers principally to the total inspired Word of God and to Jesus, Who is the living Logos.

Biblical Examples of Logos

The following passages of Scripture give examples of the logos of God:

• “In the beginning was the Word [logos], and the Word [logos] was with God, and the Word [logos] was God” (John 1:1).
• “The seed is the word [logos] of God” (Luke 8:11).
• “Holding forth the word [logos] of life” (Philippians 2:16).
• “Study to show thyself approved unto God, a workman that needeth not to be ashamed, rightly dividing the word [logos] of truth” (II Timothy 2:15).
• “For the word [logos] of God is quick, and powerful” (Hebrews 4:12).
• “Being born again, not of corruptible seed, but of incorruptible, by the word [logos] of God, which liveth and abideth forever” (I Peter 1:23).

"Rhema—The Spoken Word

The second primary Greek word that describes Scripture is rhema, which refers to a word that is spoken and means “an utterance.” A rhema is a verse or portion of Scripture that the Holy Spirit brings to our attention with application to a current situation or need for direction.

Every word of God is inspired, and “all scripture is given by inspiration of God, and is profitable for doctrine, for reproof, for correction, for instruction in righteousness” (II Timothy 3:16). It is the Holy Spirit Who illuminates particular Scriptures for application in a daily walk with the Lord.

The above is taken from the this website article.

I also don't think the difference relfects a deeper or a "mystery" type of understanding. The two words have different definitions and I don't want to "read" anything into what they mean because their meaning is pretty straight forward as far as I'm concerned.

Since this answer from 4 years ago was brought up yesterday I felt the need to add to my answer specifically dealing with John 1:1 whether or not the "Logos" is the "spoken" word of God or the "plan" of God.

You will note that the John 1:1 says "the" (definite article) word (singular) existed. It does not say God's word or words, it says "The Word existed..." And "The Word existed" how?

The Word existed with "THE" God. At John 1:1b The Word and The God are distinct, there is a subject/object distinction between the two. And finally, "The Word was God." So whatever God is, The Word is, and vice versa. So if "The Word" is impersonal, God is impersonal...If "The Word" is just a thought or a plan, then "The God" is just a thought or a plan.

And finally, "and The Word was God." So the nature of The Word is God, so whatever The God is The Word is also. We also learn from John that the Word is self existent, that the Word is eternal, that the Word exists in relation to God, and that the Word is in very nature God.

In the beginning was the Word and the Word was with the Supreme Divinity and the Word was Supreme Divinity. The Bible concerns itself with God's revelation of Himself to man. Just as words convey ideas, so the "Word/Logos" reveals who God is. This is backed up at John 14.

I think the following explains it pretty good.

What Is a “Rhema”?

Scripture for personal application

Logos—The Word of God

There are two primary Greek words that describe Scripture which are translated word in the New Testament. The first, logos, refers principally to the total inspired Word of God and to Jesus, Who is the living Logos.

Biblical Examples of Logos

The following passages of Scripture give examples of the logos of God:

• “In the beginning was the Word [logos], and the Word [logos] was with God, and the Word [logos] was God” (John 1:1).
• “The seed is the word [logos] of God” (Luke 8:11).
• “Holding forth the word [logos] of life” (Philippians 2:16).
• “Study to show thyself approved unto God, a workman that needeth not to be ashamed, rightly dividing the word [logos] of truth” (II Timothy 2:15).
• “For the word [logos] of God is quick, and powerful” (Hebrews 4:12).
• “Being born again, not of corruptible seed, but of incorruptible, by the word [logos] of God, which liveth and abideth forever” (I Peter 1:23).

"Rhema—The Spoken Word

The second primary Greek word that describes Scripture is rhema, which refers to a word that is spoken and means “an utterance.” A rhema is a verse or portion of Scripture that the Holy Spirit brings to our attention with application to a current situation or need for direction.

Every word of God is inspired, and “all scripture is given by inspiration of God, and is profitable for doctrine, for reproof, for correction, for instruction in righteousness” (II Timothy 3:16). It is the Holy Spirit Who illuminates particular Scriptures for application in a daily walk with the Lord.

The above is taken from the this website article.

I also don't think the difference relfects a deeper or a "mystery" type of understanding. The two words have different definitions and I don't want to "read" anything into what they mean because their meaning is pretty straight forward as far as I'm concerned.

Since this answer from 4 years ago was brought up yesterday I felt the need to add to my answer specifically dealing with John 1:1 whether or not the "Logos" is the "spoken" word of God or the "plan" of God.

You will note that the John 1:1 says "the" (definite article) word (singular) existed. It does not say God's word or words, it says "The Word existed..." And "The Word existed" how?

The Word existed with "THE" God. At John 1:1b The Word and The God are distinct, there is a subject/object distinction between the two. And finally, "The Word was God." So whatever God is, The Word is, and vice versa. So if "The Word" is impersonal, God is impersonal...If "The Word" is just a thought or a plan, then "The God" is just a thought or a plan.

And finally, "and The Word was God." So the nature of The Word is God, so whatever The God is The Word is also. We also learn from John that the Word is self existent, that the Word is eternal, that the Word exists in relation to God, and that the Word is in very nature God.

In the beginning was the Word and the Word was with the Supreme Divinity and the Word was Supreme Divinity. The Bible concerns itself with God's revelation of Himself to man. Just as words convey ideas, so the "Word/Logos" reveals who God is. This is backed up at John 14.