ANSWER: First, let's look at the dictionary definition of these words. The following is from the New Illustrated Webster's Dictionary (c.1992).
Worship - 1. The act or feeling of adoration or homage; the paying of religious reverence, as in prayer, praise, etc. 2. The act or feeling of deference, respect, or honor toward virtue, power, or the like. 3. Excessive or ardent admiration; also, the object of such love or admiration. 4. A title of honor in addressing persons of station.
Praise - (verb): 1. To express approval and commendation of; applaud, eulogize. 2. To express adoration of; glorify (God, etc). (noun): 1.Commendation expressed, as of a person for his virtues, or concerning meritorious actions; utterance of approval; honor given; also, applause. 2. Thanksgiving for blessings conferred; laudation to God; worship expressed in song. 3. The object, ground, reason, or subject of praise.
By comparing the two definitions, we can first see that praise is part of our worship of Almighty God. We express our approval, thankfulness, and respect for Almighty God by offering praises to Him. This kind of thankfulness then becomes part of our worship of the Great God. In our worship of God, we show our love through obeying His commandments, living the way of life He has revealed through His written word, the Bible, communicating with Him in prayer, meeting and fellowshipping with other believers, and in offering thanks to Him.
We reviewed Strong's Exhaustive Concordance of the Bible definitions in the Hebrew and Greek for both words. They are similar to what we found in Webster's. Worship means to prostrate oneself before God, bow down, humbly beseech, and do obeisance or reverence. In worship, we demonstrate that compared to God, we are as nothing. Praise means to revere (with extended hands), laudation, specifically a hymn, to kneel and to bless God.
Here are a few examples of the Biblical use of these words.
"All the ends of the world shall remember and turn to the Lord, and all the families of the nations shall worship before You." (Psalm 22:27, NKJV throughout)
"Give unto the Lord the glory due to His name; Worship the Lord in the beauty of holiness." (Psalm 29:2)
"The woman (woman Jesus met at a well in Sychar, a city of Samaria) said to Him, 'Sir, I perceive that You are a prophet. Our fathers worshiped on this mountain, and you Jews say that in Jerusalem is the place where one ought to worship.'
"Jesus said to her, 'Woman, believe Me, the hour is coming when you will neither on this mountain, nor in Jerusalem, worship the Father. You worship what you do not know; we know what we worship, for salvation is of the Jews. But the hour is coming, and now is, when the true worshipers will worship the Father in spirit and truth; for the Father is seeking such to worship Him. God is Spirit, and those who worship Him must worship in spirit and truth.' " (John 4:19-24)
"Now I, John, saw and heard these things. And when I heard and saw, I fell down to worship before the feet of the angel who showed me these things.
"Then he said to me, 'See that you do not do that. For I am your fellow servant, and of your brethren the prophets, and of those who keep the words of this book. Worship God.' " (Revelation 22:8-9)
"My praise shall be of You in the great assembly; I will pay My vows before those who fear Him." (Psalms 22:25)
"And the twenty-four elders and the four living creatures fell down and worshiped God who sat on the throne, saying, "Amen! Alleluia!" Then a voice came from the throne, saying, 'Praise our God, all you His servants and those who fear Him, both small and great!' " (Revelation 19:4-5)
In summary, praise is a form of worship, which involves the whole aspect of our awe, reverence, love, and respect for Almighty God.