The actual size of a screen is less important than the resolution, i.e. number of pixels on the screen. For instance, if you move from a 1920×1080 4.5 inch screen to 1920×1080 5.5 inch one, there are chances things will appear the way they did before, but just larger.
When choosing to buy a new device, look at the dots per inch (DPI) or pixels per inch (PPI). Higher DPI means that the pixels are smaller, or, in other words, if two screens have the same size, the one with higher DPI will have more pixels. In general, higher DPI leads to “nicer” screens, which is especially noticeable for text and vector graphics.
There could be benefits to get a larger screen with the same DPI: for instance, reading small text on a tiny screen could be very problematic; the same text on a larger screen would be more readable, even if DPI stays the same. However, lower DPI could make the text much less readable even on a larger display; for instance, after using for a year an old HTC phone at 469 ppi, I found Samsung Galaxy Tab A with its 132 ppi to be absolutely unusable for reading, even if its screen was much larger than HTC's one.
I'm unaware if it's possible to change the resolution on Android devices without rooting it, but it would be a bad idea anyway. What would happen is that a single pixel will have to display, say, what would correspond to 1.3 pixels. As a result, you will be able to see more icons and text, but it would be terrible visually.
For the same reason (as well as to reduce the impact on the GPU), most PCs do let you change the resolution, but won't allow you to set one which is superior to the native resolution of your monitor.
Another issue which appears when changing the resolution is the fact that some applications on Google Play may be marked as incompatible:
Keep in mind that altering your DPI may cause the Google Play Store to incorrectly label your device as incompatible when downloading apps. This is caused by developers limiting their app's compatibility based on a device's DPI value.