They won't be the same.
If you run a discrete simulation you will get the actual (or an instance of an actual path) price process for the future value of the stock using the real probability measure.
If you do the same thing using the closed form solution, the path will look very similar but will drift downwards.
Why are they different?
To see it easily, build a spreadsheet model with a graph that shows both the real and the modeled path (the latter being the one with $e^{r-\sigma^2/2)}$. Then plug in maybe 5% for $r$ (or $\mu$, they are the same). Then run it using $\sigma=0$ and perhaps $\sigma=40\%$.
It will be clear that with no risk ($\sigma=0$) the path is just $S_t=B_0e^{rt}$, where $B_0$ is the price of the bond at time $t=0$. It drifts up in value to return the risk free rate over a single period (a year). This makes sense.
However, with $\sigma=40\%$ the modeled price process for a stock that starts at price $B_0$ drifts downwards.
The whole point of a risk-neutral measure and model is that you discount future amounts by the risk-neutral, or risk-free, rate. It doesn't make that real, or make the stock's expected return the same as a bond. It just makes it consistent.
So imagine a stock with an initial price of $S_0$. If the stock has a higher risk than the bond (which it must) and investors in equilibrium have bid the price to a point so it is expected to have a return greater than the bond to compensate for the risk, it must be that the stock is priced a discount to the bond if investors expect the future value to be equal. Thus, if investors expect $B_{t=1}=S_{t=1}$then $S_0<B_0$. In essence, the stock is priced today at a discount to the bond.
The closed-form solution does everything in risk-neutral space. So if we start with $S_0=B_0$ the bond trajectory of price $B_t$ must discount back to $B_0$ when the risk-free rate is used. As a result the future value of the stock at the same time must be below $B_t$ so that it discounts back to a lower value at $t=0$ using $r$ as the discount rate to earn a return that compensates for the risk.
Simply, if you 'roll forward' a simulation the stock will outperform the bond on average, but if you see a price model under risk-neutrality the path must be such that when you discount future values to today they must give you a fair value today for the stock.
This is a bit of mathematical sleight of hand but it all works out the same. So, for example, if $B_0=100$ and $r=5%$ the future value of the bond in one year is 105, and its present value is 100. But the future value of the stock must look like a smaller number (say, perhaps, 94) so that the price today, $S_0$, is maybe 89 or some such.
The closed form solution does not give you the actual price model. It gives you a future price model that allows you to price a stock as if the risk-free rate can be used to discount the future value to get the right present value. They are really the same model just expressed differently.