I found PIE roots described in Etymonline (or American Heritage Dictionary) and Wiktionary are quite different. For examples:
I have no idea which one is correct or recent, due to my lack of knowledge in PIE. Why are they different (in terms of source) and where the differences come from (in terms of form)?
The main problem with these particular reconstructions is that the author of "etymonline" does not use diacritics. In fact, there is a very significant difference between *g and *ǵ (they develop differently in the “kentum” and “satem” languages), and also between *a and *ā (or, if you prefer, *h₂e and *eh₂). The reconstructions in “etymonline” are wrong, regardless of which particular school you prefer.
Having said this: all reconstructions of PIE are hypothetical and there is substantial disagreement between specialists about nearly everything.
To take one example: there is actually no direct evidence (for example from Hittite) for a laryngeal in *h₂eǵ-. It has been posited on the basis of the dogmas that no PIE word can begin with a vowel, and that there is no *a vowel in PIE. Without these two assumptions there is no strong reason not reconstruct *aǵ-. But *ag- (as mentioned) is definitely wrong.
Proto-Indo-European has gone through different stages of development historically, which represent higher levels of abstraction.
In particular, the https://en.wikipedia.org/wiki/Laryngeal_theory, which dates from 1879 but which gained widespread acceptance only after it was used to make sense of Hittite in the 1930s, did away the reconstruction of long vowels and /a/ in Indo-European, by explaining them via laryngeals (h1, h2, h3). While the reconstruction of long vowels and /a/ is legitimate, they are held to represent a later stage in the development of Indo-European. (So /h₂e, eh₂/ > /a/.) And even after the laryngeal theory gained general acceptance, references were slow to adopt it; Pokorny's influential dictionary dates from 1959, but still uses the older vowels.
Etymonline reflects the older, pre-laryngeal understanding of PIE (which corresponds to a later stage of PIE). Wiktionary reflects the more recent, laryngeal understanding. In addition, Etymonline seems to be randomly dropping diacritics, reflecting bad transcription online; the differentiation between velar g and palatal ǵ has been in the reconstructions of PIE since the start. Nothwithstanding the usual cautions about Wiki sources, Wiktionary is by and large reliable for PIE.
To supplement Nick's excellent answer:
The mainstream view of PIE now is that it had no /a/ vowel (in the oldest stages we can reconstruct). Instead, it had three(*) "laryngeal" sounds, which weren't actually particularly laryngeal, but the name has stuck. They're conventionally written *h₁ *h₂ *h₃ and there are various theories as to their actual pronunciation: personally, I like the idea that they were /h x ɣʷ/, but there's not enough evidence to be sure.
(*) Some people suggest that there were four or even five, but this hasn't caught on.
But in every Indo-European language known at the time(**), these "laryngeals" never appeared directly. Instead, they only appeared as a "color" on the surrounding vowels. In other words, *h₁ *h₂ *h₃ first affected the surrounding vowels, then disappeared entirely.
(**) Hittite was discovered after the laryngeal theory was proposed, and did actually show direct evidence of some laryngeals. Which adds quite a lot of evidence to the theory.
In particular, *h₁ turned *e into *e (no change), *h₂ turned *e into *a, and *h₃ turned *e into *o. If the laryngeal came before the vowel, the result was short; if it came after the vowel, the result was long.
So Wiktionary is showing the (more modern) reconstruction of older PIE, while the laryngeals still existed, and Etymonline is showing the (older) reconstruction of later PIE, after Hittite split off and the laryngeals vanished.
Note, as the others have said, that Etymonline is also ignoring several important distinctions in PIE: *ǵ and *g are separate phonemes in every reconstruction I've seen, and there's significantly more evidence for them than there is for laryngeals. (See the centum-satem division for examples.)
To add to all the other excellent answers, there is a simple practical explanation of the proximal cause (the others explain well the academic reasons).
Checking the sources of the two sets, there is a great amount of overlap. Both Etymonline and Wiktionary list as sources Calvert Watkins, The American Heritage Dictionary of Indo-European roots, Carl Darling Buck, A dictionary of selected synonyms in the principal Indo-European languages, and Julius Pokorny, Indogermanisches Etymologisches Wörterbuch. These three all use the Pokorny style of writing (as others have mentioned describing a more recent PIE.
But Etymonline does not and Wiktionary does list Rix et al., Lexikon der Indogermanischen Verben (1998), which uses a more modern transcription, directed at an older PIE.
And on the whole Wiktionary seems to use the latter Rix transcription.
