How do FAR 23 takeoff requirements compare to FAR 25 requirements?
The single biggest difference is that FAR 25 has rules that pertain to wet runways. Aside from that, FAR 25.113(a) uses the greater of two tests to determine accelerate-stop distance:


(1) Accelerate to $V_{EF}$, then reject the takeoff
(2) Accelerate to the highest speed required for takeoff (somewhat unclear?), then reject the takeoff


...while 23.55(a) is more or less equivalent to 25.113(a)(1).

Comparing the three methods, the rules are mostly comparable. The only major difference is that under Part 25 "a distance equivalent to 2 seconds at the $V_1$ for takeoff from a dry runway" (25.113a.1.iv and 25.113a.2.iii) must be added to the other factors. 

Because aircraft certified under Part 25 must use the greater of two different tests and add a distance equivalent to 2 seconds at $V_1$, accelerate-stop distances will likely be greater than for Part 23-certified designs that are otherwise similar.



For reference:


FAR 23.55(a)
FAR 25.113(a)