Assuming you already have a 0% interest credit card with a long interest-free period:
Main advantage with the 0% interest card is you're in more control over the repayments. So one month, you may only want to pay the statement minimum, whereas another month you may want to pay much more. So long as you're fully paid up by the time the interest-free period runs out, then you're good. One needs financial self-discipline for this approach!
Compare this with the Barclays financing, where it's a set amount each month. Making additional payments does not reduce future monthly amounts - it only shortens the repayment period. Barclays is over 24 months, whereas the 0% credit card is whatever interest-free period you've managed to obtain. Recent searches I've done for myself show very few offers longer than 16-20 months, with most options at 9-12 months. I compare this to several years ago when I was inundated with 25 - 28 month options.
Barclays will do a hard search, whereas your existing 0% card will have already had a search done against it when taking it out initially. If it's a new card, then both Barclays and the new card application will incur a hard search.