Teacher translates the word διδάσκαλος. Logos Bible Software charts the sense as only teacher.
Instructor translates the word καθηγητής. Logos Bible Software charts the senses as only tutor. However, the BDB lexicon defines it as teacher.
The terms are synonyms. However, the way διδάσκαλος is used with Rabbi. It is probably translated Jesus' original word Rabbi. A major modern Hebrew translation, translates διδάσκαλος as Rabbi in this verse. Rabbi is also translated master. Thus, for the usage in this verse teacher is apparently a higher level than instructor.
Key to your second question is "and you are all brothers." Thus, Jesus' point isn't to get rid of titles, but to get rid of rank. He is saying Christians are all siblings on an equal level. In modern day terms you could think of the teacher as the the pastor and the instructor as a deacon or Bible study teacher. Many denominations if addressing someone with a title use brother or sister, and don't address them with a title for their position.
We understand the term Rabbi, but because the Greek terms translated Hebrew/Aramaic terms unknown to us, and the Greek terms were not used for positions in the church, we can only say these are terms giving someone prominence.