I thought the exact same thing, and wrote a program to test it. Program is below:

 from random import random
p_success = 0.10
def twelve_trials(p_success = 0.25):
>>>># Runs twelve trials, counts the successes
>>>>success_count = 0
>>>>num_trials = 0
>>>>for i in range(12):
>>>>>>>>if random() < p_success:
>>>>>>>>>>>>success_count += 1
>>>>>>>>num_trials += 1
>>>>return success_count
def trials_until_3(p_success = 0.25):
>>>># Runs trials until it hits three successes, counts the trials
>>>>success_count = 0
>>>>num_trials = 0
>>>>while success_count < 3:
>>>>>>>>if random() < p_success:
>>>>>>>>>>>>success_count += 1
>>>>>>>>num_trials += 1
>>>>return num_trials
for i in range(100):
>>>>num_tests = 10000
>>>>twelve_trials_successes = 0
>>>>for i in range(num_tests):
>>>>>>>># See how often there are at least 3 successes in 12 trials
>>>>>>>>twelve_trials_successes += (twelve_trials(p_success) >= 3)
>>>>
>>>>trials_until_3_successes = 0
>>>>for i in range(num_tests):
>>>>>>>># See how often 3 successes happen in 12 trials or less
>>>>>>>>trials_until_3_successes += (trials_until_3(p_success) <= 12)
>>>>print '{0}\t{1}'.format(twelve_trials_successes, trials_until_3_successes)

Turns out they actually are equivalent. I tested with all manner of probabilities of success. Obviously, if what you're actually doing is running a set number of trials in one case and running trials until you reach significance or give up in the second case, you will come up with different results. However, if you have a set number of trials and a set success threshold set beforehand, it doesn't matter whether or not you run all the trials, or just run until the success threshold (which actually seems fairly obvious in retrospect). Edit: formatting sucks