I recently noticed that some of my cards have a black border in contrast to the white border of the new cards.Are the black bordered cards playable?What's the difference?

If they come from the Clash of Arms cycle, or I think from one of the later editions of the CCG, then they remain playable. Valyrian Edition is sticking in my mind, for some reason. The only difference is they were printed before the CCG became the LCG, as far as I can tell.

Maybe one day, FFG will reprint them with white edges, but who can say as to when that will happen?

All cards printed in the old CCG format have the black borders. They are not legal for sanctioned tournaments (aside from any sanctioned "classic" tournament formats that specifically allow them). When FFG switched to the chapter pack format, they released one chapter pack cycle, A Clash of Arms, that ended up being legal for both CCG and LCG formats. Since the cards of that cycle were designed before the transition to the LCG format took effect, they retained the black borders of the CCG era.

If you are unsure whether a particular card is from the Clash of Arms cycle, check for an "F" in front of the cards number. This indicates that it was released in the fixed collation of the chapter pack format. Any other letter preceding the number (indicating the cards rarity) means that the card is from the CCG era.

I follow the basic rule that if it has a black border then its probably good enough to be in my deck.