There is a process out there that generates a new value each month for the series you're studying. This process is unknown to you, but you're trying to figure out what it might be by looking at only 1 realization of the process, which your sample is a subset of. If the process changed over time, then your sample is really a union of two samples from 2 different processes. Think of a series of your favorite exchange rate before and after Bretton Woods, with exchange rates essentially fixed prior to 1972 and floating thereafter. In such a case, estimating a model for the entire time period, without adjusting for the change/break in the process, is tantamount to assuming that there is one and same, unchanged process generating all the values of your time series, which is an incorrect assumption. The fact that your sample suggests that it was produced by an ARIMA (1,1,1) process if you use one set of dates and ARIMA (1,1,2) if you use a different set of dates suggests that there may have been a break in the process at some point.
Did you plot the series? Did you plot the first-differenced series? If you haven't, make sure you do plot them. A stationary process has a constant mean and constant variance (among other properties), and a visual inspection of your series and of its first difference may be informative on that front. Visual inspection of the graph may also be fruitful in hypothesizing break dates, if applicable. You'll want to make sure that there is also some theoretical, intuitive reason for a break.
A break in the process would imply that the process is non-stationary. You can test for a break in the process in two ways, which differ based on whether you know when the break occurred. In the first case, if you know the break date, you'll use the Chow test, in the second one you'll use the QLR test, which essentially performs a Chow test for every single date in your dataset and picks the date with the greatest evidence for a break in the process.
One you've confirmed that your process had a break, you can reflect it in your model by adding a dummy variable (0 before the break, 1 after the break) and also interact your break date dummy variable with all the (relevant) regressors (in your case, these would be the AR and MA terms). The Chow test mentioned above is essentially an F test that the coefficient on the break date dummy and all the coefficients on the break date dummy date interacted with other regressors, are all zero. The QLR test is essentially same as the Chow test, except that it conducts an F test at the date at which it found the F-statistic to be the highest.
Because you do not know when the break occurred, use the QLR test to determine the break, unless visual inspection of the graph yielded a hypothesis for the break, supported by theory and intuition, in which case you can use the Chow test.