I have done some research, and I have a hypothesis, largely based on an answer to this same question on Quora. First, allow me to state my research. I looked up the definitions of these two words on Merriam Webster. The definitions are close if not synonymous.
1 : exciting wonder : marvelous, astonishing · a sight wonderful to behold
2 : unusually good : admirable · did a wonderful job
: that is to be marveled at : extraordinary · a wondrous feat
The first definition of wonderful is quite similar to the definition of wondrous. So I don't think we can draw a distinction just using the first sense given for wonderful. The second sense, however, is a bit different. There is a difference between something that is "extremely good" and something that excites wonder. The latter is more poetic and emotional.
This Quora answer seems to touch on this. It claims that the two words are synonymous, but that wondrous is better reserved for poetic and emotional situations.
This has led me to form a hypothesis, and I would like to know whether you think it correct. Has the word wonderful shed its first meaning and adopted the second? I am referring in particular to the meanings given within the Merriam-Webster definition.
Does wonderful have less to do with things that excite wonder, nowadays, in American English, and more to do with things that are extremely good? Does wondrous do a better job of capturing that lost meaning of wonderful, i.e. things that excite wonder?
If wonderful has indeed shed its first meaning of "exciting wonder", then I would guess this is because the word is simply too cliché. I would assume the same is true for words like amazing, awesome and awful. It would be more poignant to say that someone was in awe, or that something was awe-inspiring or mesmerizing, as these words are used less often.