Why would a nonlinear programming solver come p with a different solution than a linear programming solver if all the constraints are linear? Isn't a linear programming solver basically a "subset" of a nonlinear programming solver?
I cannot agree with the answer of Nunoxic nor the strong words of the comment of Will Jagy, so I decided to write my own answer (the comment is too short).
I see three major possible reasons why those results differ:
Ad 1. Those differences might be unnoticeable, i.e. maybe you use rational numbers in one input and floats in the other? Or you use some approximations like $1.333$ instead of $4/3$? Moreover, different algorithms, when arriving at their internal representations, may arrive at slightly different problems (for the same reasons). Finally, you might have made a mistake (e.g. the minimization function not being linear, etc.) ;-)
Ad 2. I think this is clear, if there are multiple solutions and you use two different algorithms. In fact this may even happen for the same algorithm, if it is randomized.
Ad 3. This is a huge problem and there is no cheap workaround. Best, if you can, use rational numbers in your linear programming, however, in general, you are doomed. All gradient descent methods and other numerical ways are more like heuristics, not exact solutions (imagine how easy it is for a program to miss a zero of a function if there is nothing around this place that points to it). It is hard to tell anything without symbolic manipulation, and this is often too difficult for computers. So it may happen that the algorithm, being just oblivious, decided that the solution it got to is good enough and that it won't help much by approximating it further. On the other hand, the linear programing optimizer, being not that oblivious as the previous algorithm, knows that it can approximate the exact solution if it runs long enough.
Hope that explains something ;-)
Because things like these play hide and seek. You know it's linear but the computer doesn't know that until you code it explicitly to behave differently when you have different constraints and creating such a (I'll call it) decision tree will ruin the coder's life. Usually, the rule is that the programmer's time is more important than the system run time and thus, you'd probably expect that linear systems tend to take same time as nonlinear systems because the underlying mechanism for solving might be same. Same goes with answers, you'd expect different answers because although you know that the constraints are linear, unless you communicate it to the computer in some form, it is bound to assume otherwise and apply a generic algorithm.
The tradeoff between the complexity of your code and its performance is something all coders look at. A very good example of this would be MATLAB's
On a tangential topic (but something I thought worth mentioning) is a similar question Mathematicians have with respect to high performance computing (Super computing included). The common complaint is "Processes X and Y happen independent of each other, why can't the computer (compiler) realize this and automatically make it run on multiple cores". The answer here is the same: The coder and the compiler play hide and seek with such optimizations. Unless the coder sits and explicitly codes the math at a very low level (maybe even ASM), it's unlikely that you'll extract the largest punch from the supercomputer.