Please look up the original (Greek/Hebrew) words for fear in Proverbs 8:13 and 1 John 4:18. I feel sure that the passage in Proverbs and other passages in the Bible that say to fear the Lord are the same ("to hate evil"), and that the other passage is a different word meaning "to be afraid of" or "mistrust." My conclusion is that as our love for the Lord grows (love=to keep his commandments), then our fear of the Lord would also grow (fear=to hate evil).
Answer
In the Hebrew there are four words designating fear. Two are frequently translated "fear" or "dread" or "terror." "And the fear [mowra'] of you and the dread [chath] of you shall be upon every beast of the earth." (Gen 9:2) "the LORD your God shall lay the fear [pachad] of you and the dread [mowra'] of you upon all the land." (Deut 11:25) The word "chath," translated "dread" in Gen 9:2 also has the meaning "broken" in 1 Sam 2:4: "The bows of the mighty men are broken." Another such word that in context may be either "fear," "break," or "prevail" is "arats." It is used in the context of fearing God in Isaiah 29:23 and Isaiah 8:12-13.
Three of the four words are used in the sense of "fear God." "Mowra'" is used this way in Psalm 76:11, Malachi 1:6, and Malachi 2:5. I have already mentioned such a use of "arats." By far the most common word in Hebrew used for "fear God" is "yare" or its variations. It is used in the sense of being afraid and "fear God" in the same verse in Ex 20:20-"And Moses said unto the people, Fear not: for God is come to prove you, and that his fear may be before your faces, that ye sin not." This makes understanding what is meant by the fear of God more difficult, because he says here, "don't be afraid, but fear God," and both fears are the same. The word is also translated "terror," and, as a lesser meaning, "awe or reverence." Putting it in a family perspective, the same word could be used for a child who "fears" an abusive parent and a child who "reverences" a non-abusive parent who may occasionally reasonably punish.
The Jews have another perspective on the idea of "fear God." While it is commanded of a Jew, the term "a God fearer" is generally used of a gentile who, while not converting to Judaism, acknowledges the one God of the Jews. This is the sense Paul uses in Acts 13:16, when he spoke in the synagogue in Antioch Pisidia. It was, in this sense, also said of Cornelius in Acts 10:2.
In the King James Version of the New Testament, two Greek words are translated "fear." The word "deilia" is used only once, in 2 Tim 1:7, and is more accurately translated "timidity." All other uses of the word are translated from variants of the root word "phobos," from which we get phobia, an unreasoned terror of something. This is the word used in Matthew 14:26 when the disciples saw Jesus walking on the water and "cried out in fear." The same word is used in Acts 9:31, "Then had the churches rest throughout all Judaea and Galilee and Samaria, and were edified; and walking in the fear of the Lord, and in the comfort of the Holy Ghost, were multiplied." Here it joins the fear of the Lord and the comfort of the Holy Spirit. It is also the same word used in the passage you mentioned, 1 John 4:18, "There is no fear in love; but perfect love casteth out fear: because fear hath torment. He that feareth is not made perfect in love."
So in answer to your comment, there is no different word used in 1 John 4:18. How, then, do we reconcile the two concepts?
Israel was told that God required them to fear Him. And now, Israel, what doth the LORD thy God require of thee, but to fear the LORD thy God, to walk in all his ways, and to love him, and to serve the LORD thy God with all thy heart and with all thy soul, To keep the commandments of the LORD, and his statutes, which I command thee this day for thy good?" (Deut 10:12-13) In fact, the only thing in God's treasure house is "the fear of the Lord." (Isa 33:6) The church was commended for walking in the fear of the Lord (Acts 9:31, above). So how to reconcile that with perfect love casting out fear?
I think we need to look at the context of 1 John 4:18. I think verses 14-21 will suffice.
And we have seen and do testify that the Father sent the Son to be the Saviour of the world. Whosoever shall confess that Jesus is the Son of God, God dwelleth in him, and he in God. And we have known and believed the love that God hath to us. God is love; and he that dwelleth in love dwelleth in God, and God in him.
Herein is our love made perfect, that we may have boldness in the day of judgment: because as he is, so are we in this world. There is no fear in love; but perfect love casteth out fear: because fear hath torment. He that feareth is not made perfect in love. We love him, because he first loved us. If a man say, I love God, and hateth his brother, he is a liar: for he that loveth not his brother whom he hath seen, how can he love God whom he hath not seen? And this commandment have we from him, That he who loveth God love his brother also.
What is "perfect love" or "completed love?" John says it is that we may have boldness in the day of judgement. In the context above he is saying that when the judgement comes, we will know that Jesus is the savior and that knowledge will enable us to be bold before the judgement seat of God, because we are love just as God is love. He concludes with a test by which we can know that God dwells in us in love: if we don't love others we can know we don't love God. If we don't love others, we should be terrified rather than bold in the day of judgement. But if we love, we can be bold. Therefore, the casting out of fear here is not in the same context as fearing God. We walk in the fear (combined terror/reverence) of God, but we do so boldly. Perhaps it is a bit of a paradox, or even an oxymoron (bold fear). Maybe we won't fully understand it until we experience the judgement, but we must fear God so that we won't fear God.