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_{^{Triangulating for the simplest puzzle that is still at least somewhat interesting to solve..}}

On the left side wall in this picture, we have two particular clocks:

1: an analog clock with identical hour and minute arms, and

2: a digital clock that shows initial zeroes when appropriate

During the course of a day, which of these clocks agrees more often with its horizontal mirror image?

The times where the analog clock will match are

The same as the times that the analog clock has reflective symmetry across it's "y" axis. This will happen at least once an hour. The proof for this involves the Intermediate Value Theorem, but essentially boils down to the fact that, every hour, the hour hand continuously sweeps one part of the clock, while the minute hand continuously sweeps the entire clock. It follows that, in sweeping the entire clock, it must for an instant pass through the part of the clock that is currently opposite the hour hand.


It actually matches slightly more often than this, because they're already matching at noon/midnight, so it comes out to 13 matches every 12 hours, or 26 total.

The times where the digital clocks will match are

Again, where the clock has reflective symmetry across it's "y" axis (The colon). As @Dorrulf has already enumerated, this happens at 00:00, 02:50, 20:05, and 22:55. I'm not counting the times with ones in them, because they don't show up on the same half of the digit when reflected. This is a measly 4 total

So, it's clear that the reflections match more often for the

Analog Clock

Last try...

As a forward, I'm going more for the idea that the clocks read identical times rather than they just look similar.
The only analog positions I see matching would be at 12:30 and 3:45, and their self-reflections (6:00 and 9:15) for a total of 8 occurrences in 24-hr a day. (Thanks @ExcitedRaichu)

The digital clock has a few things of note beforehand:

The leading 0's and set to military time.

Again, if we function with the idea that as long as the times read the same (and the positions don't have to be exact - left oriented 1 versus right oriented 1), then we have these reflective cases: 0 to 0, 1 to 1, 2 to 5, 5 to 2. We can't use 8 because minutes only goes to 60. This gives us the following combinations (barring any were missed): 01:10, 11:11, 10:01, 15:21, 12:51, 05:20, 02:50. @DqwertC also found: 00:00, 02:50, 20:05, and 22:55.

That puts us at a ration of 8:11, in favor of the digital clock.

Assuming 'agrees with' is synonymous with 'looks the same'...

Analog

As suggested by DqwertyC, the analogue clock will agree 26 times in 24 hours if you include both mid-nights. If you argue that there is only one midnight in a 24 hour day then this can be reduced to 25 times. Assuming the latter these times are approximately: 00:55, 1:50, 2:46, 3:41, 4:37, 5:32, 6:28, 7:22, 8:18, 9:14, 10:09, 11:04, and 12:00 plus the equivalent times in the afternoon.

Digital

As suggested by Dorrulf and DqwertyC there are a minimum of 4 times when the digital clocks agree (if only (2|5), and (0|0) digits are considered to be mirror images, and a maximum of 11 times if (1|1) is also included.

Conclusion

Either way, 12 (or 13) trumps 11 (or 4), so the analog clock wins.

Analog clock matches are:

One math per hour (0 to 11 inclusively) = 12 matches
An additional match at 6:00 = 1 match
Those matches occurs twice per day since the hour handle makes 2 complete turns by day.
(12 + 1) × 2 = 26 matches.

For the digital clock we have:

The following symmetries : 0 <--> 0, 1 <--> 1, 2 <--> 5 and 8 <--> 8
Starting at midnight, we have:
Leading 0: 00:00, 01:10, 02:50, 05:20 = 4 matches
Leading 1: 10:01, 11:11, 12:50, 15:20 = 4 matches
Leading 2: 20:05, 21:15, 22:55 = 3 matches
Leading 5: No matches
Leading 8: No matches
Thus we have 11 matches per day

So the result is:

The analog clock matches more often.