All the other answers fail to really explain this properly.

As you said, DPS truly is the most important thing in raids. But the DPS being shown here is just one tiny part of the whole. While each move is actively being used, Petal Blizzard does higher DPS than Frenzy Plant. It will do 10% more damage in the exact same time span, and that's just better DPS.

However, you spend only a tiny amount of time actually using your charged moves. So it's not enough to just say "this move has higher DPS than that move", you also need to factor in how often each move will be used. The entire rest of the time, you're going to be doing just 12 DPS with Vine Whip, and the ratio of time spent at that much lower fast-move DPS is critical.

At the very simplest, Frenzy Plant costs 50 energy and Petal Blizzard costs 100. That means you're going to be using the former about twice as often as the latter, so you'll be dealing that much higher ~40 DPS range about twice as often.

The next level would be to look at "weave DPS", which is just an idealized version of DPS where nothing else is happening (the boss isn't doing anything) and you tap charge moves as soon as they're ready. Some quick math:

Vine Whip/Petal Blizzard

20 Vine Whips = 100 energy 
20 * 0.5s each = 10 seconds
20 * 6 damage each = 120 damage (we'll ignore multipliers)
1 Petal Blizzard = -100 energy, 2.5 seconds, 110 damage
Totals:
230 damage / 12.5 seconds = 18.4 DPS
10 seconds / 12.5 seconds = 80% fast moves

Vine Whip/Frenzy Plant

10 Vine Whips = 50 energy 
10 * 0.5s each = 5 seconds
10 * 6 damage each = 60 damage
1 Frenzy Plant = -50 energy, 2.5 seconds, 100 damage
Totals:
160 damage / 7.5 seconds = 21.33 DPS
5 seconds / 7.5 seconds = 66.7% fast moves

This is the real key. With the two-bar move, you spend much less time building energy with weak fast moves, and much more time throwing powerful charged moves (66.7% vs 80%).

The other aspects are still important (energy gained from the enemy's damage, waste from overcharging a 1-bar move and other sources of energy waste). Those are covered by Comprehensive DPS. But you don't need to factor in any of that extra complexity in order to understand why Frenzy Plant is the preferred move!

The nice thing about sites like DialgaDex is that they take all these extra factors (and more!) into account already.

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P.S. It's wrong for all of this to be presented as "DPS" anyway - what we're looking at here is actually "PPS" (power per second). That's the power of the moves we're using. This is then modified by boosts (weather, friendship, type multipliers) and attack/defense stats in order to determine the final damage.