My original code was just looking for the smallest; it did not properly account for all the conditions.

I corrected it and removed the break statement so now it finds the same 69 solutions (of 8-digit numbers) as Charlie found.

At first I was forcing the smaller 8-digit prime to have no repeat digits, but there is at least one solution that ends in a double 9, so I modified to account for that.

Ran in about 4 seconds.

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for i in range(10000009,100000000,10):
    s = str(i)
    if '1' in s:
        continue
    if s[6] == '9':
        setlength = 7
    else:
        setlength = 8
    if len(set(s)) != setlength:
        continue
    if str(int(s[6])+1) in s:
        continue
    if isprime(i) and isprime(i+2):
        if len(set(str(i)+str(i+2)))==10:
            print('({}, {})'.format(i, i+2))