You should know, though, that paradise and Heaven are two different places, and paradise is not for all eternity. I have heard, however, that there is a mosh pit, and the line to get in is murder.
Far be it from me to question our Lord and Savior, but this piqued my curiosity because, while I was aware that "paradise" could designate the Garden of Eden (e.g., Milton's Paradise Lost), I was pretty sure that Jesus' remark to the penitent criminal intended "paradise" (Gk. paradeisos) to be a synonym for "heaven."
A bit of online research reveals that the meanings of the words "paradise" and "heaven" are not universally agreed upon. Here, for example, is a page listing sources claiming that "heaven" and "paradise" can be understood as identical or as referring to different things:
1. often Paradise The Garden of Eden.
1. The abode of righteous souls after death; heaven.
2. An intermediate resting place for righteous souls awaiting the Resurrection.
3. A place of ideal beauty or loveliness.
4. A state of delight.
[Middle English paradis, from Old French, from Late Latin paradisus, from Greek paradeisos, garden, enclosed park, paradise, from Avestan pairidaeza-, enclosure, park : pairi-, around + daezo, wall.]
On that same page are other references implying that the word's meaning has changed over time, making the question of whether paradise cannot mean heaven somewhat unclear.
And check this reference out. In it, paradise is considered part of heaven:
Paradise. What does scripture mean by the phrase “bosom of Abraham” in the parable of Lazarus and the rich man? It appears that the answer is Paradise since the Holy Spirit tells us that Paradise is either in heaven or is heaven (2 Cor. 12:2-4),
I know a man in Christ who fourteen years ago - whether in the body I do not know, or out of the body I do not know, God knows - such a man was caught up to the third heaven. And I know how such a man - whether in the body or apart from the body I do not know, God knows - was caught up into Paradise, and heard inexpressible words, which a man is not permitted to speak. (NASB) 2 Corinthians 12:2-4
Paradise is in the third heaven. The first heaven is our atmosphere; the second heaven is our universe and the third heaven is where God lives. The third heaven is a spiritual realm. This is where Jesus promised the thief on the cross he would go. The thief believed in Jesus, and as a result, the thief did not have to do anything.
And He said to him, “Truly I say to you, today you shall be with Me in Paradise.” (NASB) Luke 23:43
There was no time, no opportunity. He just believed, depended on Jesus, and as a result he went to Paradise (Rev. 2:7).
But wait-- there's more! This is from a much more reliable source, the Catholic Encyclopedia (paragraph breaks added for easier reading):
The uncertainty and confusion of the current Jewish ideas concerning paradise may explain the paucity of reference to it in the New Testament. The first mention of the word occurs in Luke, xxiii, 43, where Jesus on the cross says to the penitent thief: "Amen I say to thee, this day thou shalt be with me in paradise."
According to the prevailing interpretation of Catholic theologians and commentators, paradise in this instance is used as a synonym for the heaven of the blessed to which the thief would accompany the Saviour, together with the souls of the righteous of the Old Law who were awaiting the coming of the Redeemer.
In II Corinthians (xii, 4) St. Paul describing one of his ecstasies tells his readers that he was "caught up into paradise." Here the term seems to indicate plainly the heavenly state or abode of the blessed implying possibly a glimpse of the beatific vision. The reference cannot be to any form of terrestrial paradise, especially when we consider the parallel expression in verse 2, where relating a similar experience he says he was "caught up to the third heaven."
The third and last mention of paradise in the New Testament occurs in the Apocalypse (ii, 7), where St. John, receiving in vision a Divine message for the "angel of the church of Ephesus," hears these words: "To him that overcometh, I will give to eat of the tree of life, which is in the paradise of my God." In this passage the word is plainly used to designate the heavenly kingdom, though the imagery is borrowed from the description of the primeval Garden of Eden in the Book of Genesis.
Another site says:
Question: "What is paradise? Is it different than heaven? Where do people go when they die until Christ comes back?"
Answer: The word paradise is used as a synonym for Heaven (2 Corinthians 12:4; Revelation 2:7). What we do know for sure is that there has always been a separation of believers and unbelievers (Luke 16:19-31). The righteous have always gone to paradise, the wicked have always gone to Hell (Hades). For right now, both heaven (Paradise) and Hell (Sheol) are “temporary holding places” until the day when Jesus Christ comes back to judge the world based on whether or not they have believed in Him. One day, all will be sent to their eternal destination. The wicked to the lake of fire (Revelation 20:11-15) and the righteous to a new heaven and a new earth (Revelation 21-22).
Earlier in the day, while researching this question at the office (shhhh), I found a source that very explicitly said that, in Christian theology, paradise and heaven were in no way the same thing. I've been trying to find that particular source again, and would gladly quote it now if I had the link handy.
Suffice it to say that paradise and heaven are seen as synonymous by a number of theologians. This doesn't answer the objective question of whether either really exists, and if they exist, whether they are identical. Perhaps they aren't.