I do not think that GPS has a form of background syntax checking like Eclipse does, at least I haven't found it.
I wrote a small sized terminal program in GPS as my first Ada program and my opinion of it is generally negative.
The browser is very confusing and it is hard to find items you are looking for.
The editor has the nasty habit of adding a column to the left and removing it dynamically based on the item under your mouse. What that does is when you are trying to select text with the mouse, your text switches right and left by one character as you move which makes precise selection a difficult thing to do
Getting in the debugger involves too many steps and it generally moves your files in the editor to the EOF position when you set breakpoints, so you will do a lot of scrolling to go back to the sections of interest.
The debugger output looks fancy with all the graphical elements but it is also cumbersome to operate.
GPS offers better help features for Ada libraries though, it's generally easier to browse the standard libraries and check them out than in Eclipse.
All in all, if Eclipse works for you, you might want to stick with it or perhaps open both on the same project and switch to GPS for things like help etc.
I had a related question out here which is on hold now (as I expect your question will soon be due to it's broad content) but I did some research and I am generally dissatisfied with what is out there in the Ada field. The language is great but it suffocates under bad tools.
One last tip: You might want to try Slick Edit as it's text editor and browsing features are quite good, although they too have flaws with Ada (for instance the beautifier).
Good luck with your venture