Two almost identical ODEs with different solutions

I'm trying to use ode15s to solve a kinetics problem involving a system of differential equations. I have written the code in two ways, both of which give the same output for a large range of inputs. (By same, I mean that the difference between outputs is on the order of 10^-22, while the inputs vary between 10^-6 and 10^-12.

However, when I use the ODE solver, the solutions are immediately very different.

Does the solution given by the matlab solvers depend on the design of the functions, or only on the input and output, treating the system of equations as a black box?

The solver just calls your function - i.e. it is a black box to MATLAB. Are you able to show us your examples?