What "more" does differencing (d>0) do in ARIMA than detrend? I recently read a discussion about ARIMA models where someone said (referring to d as in ARIMA (p, d, q)):

Its true that d=1 takes out deterministic trends when they are present
  (they would appear only in the drift term.) But it does more than
  that.

I know that's not much context, but I seem to remember reading something similar in regards to detrending via differencing.
Two questions:


*

*Does differencing (not just in an ARIMA context) do something more to your data than just detrend it? If so, what else does it do? (Add or remove?)

*There are other detrending methods, such as fitting a curve (loess, linear regression) and using the residuals as detrended data. Would these other methods not do the "more than that" that differencing does? (Hence, might they be preferrable?)
 A: Differencing isn't actually the preferred way of removing a trend---detrending is. Detrending involves estimating the trend and calculating the deviation from the estimated trend in any particular period.
The main use of differencing is to remove the problem of unit roots. A unit root arises, for example, when $\rho=1$ in the simple AR(1) model $y_{t} = \rho y_{t-1} + \nu_t$. In this case, differencing yields a stationary white noise process $\nu_t$ that is appropriate for analysis.
Differencing a process without a unit root, but with a trend, can actually produce bad results (the new, differenced error term can have a strange distribution that contains autocorrelation, but of a tricky process). Similarly, detrending a process without a trend, but with a unit root can fail to eliminate the problem of non-stationarity (that is, it doesn't fix the unit root problem).
A: Unnecessary differencing or filtering can inject structure (see Slutsky Effect:  http://mathworld.wolfram.com/Slutzky-YuleEffect.html,   https://www.minneapolisfed.org/publications/the-region/the-meaning-of-slutsky,   https://blog.minitab.com/blog/understanding-statistics/the-ghost-pattern-a-haunting-cautionary-tale-about-moving-averages,   http://www.sef.hku.hk/~wsuen/ls/immortal/y2c.html) . Sometimes a series can have a shift in the mean causing "non-statioanarity" ... the correct remedy is to neither difference or de-trend but to "de-mean" or use a Level Shift variable/filter to render the residual series stationary.  Sometimes there is more than 1 trend requiring a number of trend variables/filters ... none of which have to start at the beginning if the series. Analysis will tell you which of these three approaches


*

*differencing

*de-meaning

*de-trending


are suitable for your data.
