Is there any continuity between both stages of existence? Or does resurrection mean a complete new begin on a completely different level?
In human terms, Yes.
The Bible refrains from providing almost any information about life after death other than the all important destination of heaven or hell and how to ensure you arrive at heaven. However, heaven is in most senses just the restoration of the life originally lived by Adam and Eve before they sinned. This naturally means that unless otherwise instructed on some detail we are to imagine heaven like that. This means that unless otherwise told specifically that our memory will be erased, or that we will no longer have eyes, or that we will no longer eat food, or that we will no longer walk with two legs, laugh, jump, talk, or whatever else Adam and Eve would have done, we must assume it is unchanged. We can either say we don't know or we can say it is natural to expect no change. More than this has no biblical basis and is unfounded speculation and unhealthy curiosity at best.
Furthermore although not directly stating certain details about heaven there are many allusions to the life of heaven being similar to life on earth, as well as a continued connection of those in heaven to those on earth. For example, the Saints in heaven are observing God's judgments on earth in Revelation 19. Not everything in Revelation is to be taken literally of course, but the allusion maintains a thread between the church in heaven and the church militant on earth, and the connection is understood as strong and uniform in terms of sharing the experience of God's glory. It would be potentially misleading to make these strong allusions if at the same time the reality was drastically different.
The basic idea here is that there is no biblical basis to assert that saints in heaven are not aware of what is going on in the earth. God's glory is tied up with what is happening here on earth and it is counter intuitive to imagine he would hide the knowledge of that glory to those in heaven. Erasing our memory of his glory in our lives after we enter heaven is also very counter intuitive. To imagine that we loose any basic human powers in heaven is counter intuitive and contrary to any allusion on the subject in scripture or the Eden example. Furthermore, it is also odd to think that while God manifested his glory in our salvation on earth so that we learn to have faith love and hope and become more like Christ, that once in heaven when we finally see Christ 'face to face' he would put a bag over our eyes so that our faith, hope and love enflamed by a view of his glory in our human experience does not practically continue. It is 'our' faith, love and hope that passes into heaven and these qualities can't remain without some original 'childish' knowledge.
The impression I get from these verses is that we will be more aware than we are now, we will literally look upon the human face of Christ with our human face, and we will know not only what we know now but more fully. An adult does not loose all memory of his childish thoughts when he ages, he just sees much more than he did when he was a child. Heaven is not a strange wonderful lobotomy of the brain, or other human destruction but a restoration of the whole human life into a renewed garden of Eden with eternal communion with the saints and with God.