Okay, let's use evidence.
Anecdotal, but numerical.
The current Mac Mini is supposed to pull between 10W (at idle) and 85W (at load---this is the limit of what the PSU can take from the wall.[0]
Let's be generous and say that the Mac Mini, if left on continuously without sleeping, pulls on average 20W.
Let's be pessimistic and assume that power costs you $0.20 per kWh.
# The time in which the Mini will use 1 kWh
1000 Wh / 20 W = 50 hr
# The number of hours in one year
365 days * 24 hrs = 8760 hrs
# The number of kWh the Mini will use in one year
8760 hrs / 50 hrs = 175.2
# Cost of running the Mini continuously for one year
175.2 kWh * $0.20 per kWh = $35.04
That's amazing, no? Let's say, then, that you bought a mini for $599 dollars; your cost of ownership would be 599 + 35.04n, where n is the number of years you've owned it.
No doubt impressive.
A desktop I built relatively recently, which a Core i5 and an unnecessary graphics card, along with a few hard drives, pulls roughly 60W at idle and 150W at load (measured at the outlet by a Kill-a-Watt).
Let's be not-so-generous and say my machine pulls 100W on average, is left on continuously, and my electricity costs $0.20 / kWh.
My computer will use a kWh in 10 hours (1000 / 100); and will cost $175.20 per year to operate ((8760 / 10) * 0.2).
My initial cost for this machine (excluding monitor, keyboard, etc.) was $450. SO my cost of ownership is 450 + 175.2n.
The cost of ownership for east will equal in :
599 + 35.04n = 450 + 175.2n
149 = (175.2 - 35.04)n
149 / 140.16 = n
n = 1.06 years
So after a year of ownership the Mini becomes "worth it," if power consumption if your first requirement.
Note however, that I've given the Mini every benefit of the doubt possible, did not take significant steps to minimize the power consumption of my machine, am able to upgrade my machine piecemeal instead of discretely (and thus have significantly lowered future costs of ownership), and have assumed we've been leaving the machines on continuously.
The Mac Mini, let's say, pulls 0W when in S3 sleep.
My machine pulls 1W in S3 sleep, and about 0.2W when off (to answer your query, it wakes reliably and quickly).
If we assume my machine is asleep half of the time:
Half the time (8760 / 2 = 4380 hrs), my machine pulls 1W.
While asleep, it will take 1000 hrs to use 1 kWh.
Therefore, it uses 4.38 kWh in the time it is asleep.
Which means the new yearly cost is:
(175.2 / 2) + (4.38 * .2) = 87.6 + 8.76 = 96.36
The Mini ends up costing $17.50 yearly.
So:
599 + 17.5n = 450 + 96.36n
149 = 78.86n
n = 1.889 years
Once you factor in the cost of replacing the Mini versus upgrading a small bit of my machine, I think the answer is much less clear cut.
P.S. I don't mean to pick on you, but excitement about power consumption that doesn't factor in the numbers is a pet peeve. The Mac Mini is only "worth" it if having the computer you actually want is worth less that $80 or so per year to you.
[0]: https://support.apple.com/kb/HT3468
Edit: formatting.