Are Paradise and Heaven the same?
First consider dictionary definitions...
Heaven - The above of God, the angels, and the spirits of the righteous after death; the place or state of existence of the blessed after the mortal life.
Paradise - An intermediate place for the departed souls of the righteous awaiting resurrection... A place of extreme beauty, delight, or happiness... A state of supreme happiness; bliss.
The dictionary definitions give no biblical references so we have no knowledge of how they arrived at those definitions!
What do bible scholars say about Heaven and Paradise...
(1) Some say there is no difference.
(2) Some say Paradise is used as a synonym for Heaven.
(3) Some say that Paradise was a part of "Hades" until Jesus died on the cross... then, when his blood was sprinkled on the alter in Heaven... at that time Paradise was moved to Heaven and believers who died no longer had to wait in "Hades" to be with the Lord.  And some go further saying that when the "new earth" is created, the "new earth" will be Paradise.
The explanation that scholars give, to justify their belief in option 3, is the story of Lazarus and the rich man (read Luke 16:19-31)... before Jesus death on the cross no one could enter Heaven, so Lazarus the beggar went to "Paradise" which was part of "Hades" and the rich man went to a separate part of "Hades"... the rich man saw Lazarus in the far distance with Abraham and shouted to Abraham... Abraham responded and explained the chasm between them which no one could cross.

Biblical References...
Luke 16:19-31... "Now there was a rich man, and he habitually dressed in purple and fine linen, enjoying himself in splendor every day.  And a poor man named Lazarus was laid at his gate, covered with sores, and longing to be fed from the scraps which fell from the rich man’s table; not only that, the dogs also were coming and licking his sores.  Now it happened that the poor man died and was carried away by the angels to Abraham’s arms; and the rich man also died and was buried.  And in Hades he raised his eyes, being in torment, and *saw Abraham far away and Lazarus in his arms.  And he cried out and said, ‘Father Abraham, have mercy on me and send Lazarus, so that he may dip the tip of his finger in water and cool off my tongue, for I am in agony in this flame.’  But Abraham said, ‘Child, remember that during your life you received your good things, and likewise Lazarus bad things; but now he is being comforted here, and you are in agony.  And besides all this, between us and you a great chasm has been set, so that those who want to go over from here to you will not be able, nor will any people cross over from there to us.’  And he said, ‘Then I request of you, father, that you send him to my father’s house— for I have five brothers—in order that he may warn them, so that they will not come to this place of torment as well.’  But Abraham *said, ‘They have Moses and the Prophets; let them hear them.’  But he said, ‘No, father Abraham, but if someone goes to them from the dead, they will repent!’  But he said to him, ‘If they do not listen to Moses and the Prophets, they will not be persuaded even if someone rises from the dead.”
Matthew 12:40... "for just as Jonah was in the stomach of the sea monster for three days and three nights, so will the Son of Man be in the heart of the earth for three days and three nights."
Luke 23:43... "And He said to him, “Truly I say to you, today you will be with Me in Paradise.”
2 Corinthians 12:4... "was caught up into Paradise and heard inexpressible words, which a man is not permitted to speak."
Revelation 2:7... "The one who has an ear, let him hear what the Spirit says to the churches.  To the one who overcomes, I will grant to eat from the tree of life, which is in the Paradise of God.’
Revelation 22:2... "in the middle of its street.  On either side of the river was the tree of life, bearing twelve kinds of fruit, yielding its fruit every month; and the leaves of the tree were for the healing of the nations."

Discussion thoughts... The dictionary definitions are quite good... especially the "Paradise" definitions.  Most of the "Heaven" definition is OK but I disagree with use of the word "spirits" (my belief is that the bible is quite clear in that we will have heavenly bodies... not be just a spirit).
Regarding what the scholars proclaim... I believe the 3rd option listed has it correct.
So what do the scriptures say... Matthew 12:40 explains it perfectly when Jesus said "I will be in the heart (most bible versions use the word "heart", other versions use "deep", "depths", "ground", "grave", "belly") of the earth for 3 days and 3 nights" (we don't know where that is and it really does not matter).  What we do know is that when Jesus was dying on the cross and he told the thief, who was dying next to him, Luke 23:43 "today you will be with me in Paradise"... the "heart" of the earth (Paradise) is where they went!  They "DID NOT" and "COULD NOT" go to Heaven because Jesus had not yet ascended to Heaven until 3 days after his death!  Thus we have the difference between "Paradise" and "Heaven".  2 Corinthians 12:4 gives us another clue... Paul said he was caught up into "Paradise"... so "Paradise" after the resurrection of Jesus was in Heaven!  Revelation 2:7 provides another clue... "everyone who is victorious will eat from the tree of life in the "Paradise" of God"... this "Paradise" may be in Heaven or after the resurrection in the "new earth"!  Revelation 22:2 another clue... now the "tree of life" is in the "new earth"... John said that the "tree of life" is in "Paradise"!

My Beliefs... Another difficult subject... my personal beliefs are that "Paradise" prior to the resurrection of Jesus was in the "heart of the earth" as explained above in Luke 16, Matthew 12 and Luke 23.  After the resurrection of Jesus, and currently, "Paradise" is in Heaven (where Paul experienced "Paradise" in 2 Corinthians), and eventually at the final resurrection when a "new earth" is created, the "new earth" will be "Paradise" .
Stated a bit differently, I believe Paradise was first in the "heart of the earth", then was moved to Heaven when Jesus ascended to Heaven... during that first period, and currently, Paradise was and is that wonderful place where "believers" reside with Abraham and others, while waiting for the 1st resurrection.  After the final (2nd) resurrection, Paradise will be the "new earth" and Heaven will be the "New Jerusalem" (see the study "What is the 1st and 2nd Resurrection").