The main difference is related to the absorption of the different radiation energies. If you would only change the collection angle of the detector but not the sample tilt the way the radiation has pass the sample is changing. The lower the energy the higher the absorption of the components. If you change the tilt of the sample (with respect to the detector but also to the electron beam) also the distribution of the X-rays in your sample is changing so that also this has to be to taken into account. Therefore the tilt angle is important for quantitative analysis. The peak positions does not change at all, only the intensity is adapting by a specific proportional function.
... and if your sample surface is not flat but has pronounced facets those facing towards the detector will contribute more to the integral signal compared to those facing away.
Some other, perhaps less important changes include:
Tilting the sample with respect to the electron probe can change the depth the electron probe can penetrate into the sample and, therefore, the depth from which sample characteristic x-rays can emanate. Not much affect at small tilt, but larger affects with large tilt. Tilting the sample, generally in the direction of the detector, can greatly improve the signal and the signal-to-noise ratio. This will be most apparent with a perfectly flat sample, and was alluded to by T Walther (previous answer).
These answers are based upon electron induced EDS. Another form of “EDS” is X-ray induced, or X-ray Fluorescence (XRF). A very interesting form of the later sometimes used in connection with an SEM is micro-XRF. A focus X-ray probe is used to induce sample X-ray emission (usually a 10-50 micrometer X-ray probe is used). This has certain advantages such as higher elemental resolution from about 2KeV and above (over that of electron induced EDS), and a much reduced background or noise signal.