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These words share the Greek root πτέρυξ (pteryx), meaning feather/wing, but the P in pterodactyl is silent (in the initial position), while the P in archeopteryx (in the middle of the word) is voiced.
Why is this, and what's the rule? Is it just that that's how the ancient Greeks pronounced it, or is there something more subtle going on?
Π (pi) in Greek is pronounced as P in English, and Τ (tau) in Greek is pronounced as T in English.
Greek though is quite okay with PT at the start of a syllable, while that isn't a phoneme cluster we have in English.
So, while we can pronounce both letters if we end one syllable with P and start the next with T, we will make the P silent at the start of a word.
This has happened for some time, since while pterodactyl is a relatively recent word, we've been talking about Ptolemy for quite some time.
Comparably, when we import Ξ from Greek as X, we'll pronounce it KS in the middle of a word, but change to a Z sound at the beginning, as discussed at this question.
First, the English rule has to do with syllable structure. As Colin points out, English syllables can't begin with the consonant cluster /pt/. Any word spelled that way will be pronounced some other way.
But the sequence /pt/ can occur between syllables. So if the vowel before /pt/ is stressed (as in archeopteryx /ar.ki.'ap.tər.ɪks/), then both stops can be pronounced, each in its own syllable, as Jon Hanna has noted. That's the English rule.
Of course, both words are made using the Greek word πτέρυξ (originally pronounced /'ptɛruks/) 'wing', and this word, as well as the English words feather (p -> f and t -> θ via Grimm's Law), compete, perpetual, ptomaine, symptom, and hippopotamus, all are descended from the PIE root *pet- 'to rush, fly'. The /p/ and the /t/ were both pronounced in Greek.
Finally, the English word helicopter is also transparently derived from helico- 'rotating, spiral' plus -pter- 'wing'. What's interesting about this is that helicopter is such a common word that it's been split into combining forms, but not the original ones.
In English, helicopter is divided into heli- (heliport, helipad, helitaxi, helitours, ...) and -copter (cargo copter, minicopter, jetcopter, ...), because the glue between helico- and -pter- wouldn't come off, but the glue between heli- and -copter turned out to be soluble in English phonology.
The trick is to get the two stops into separate syllables. With archaeopteryx, the p ends one syllable and the t starts the next one. With pterodactyl, no such luxury is possible, so the p is lost.
See also apnea versus pneumonia.
No, it is not how the ancient Greeks pronounced it: /pt/ was a possible initial cluster (as were /pn/, /ps/ and /kt/).
In English, /ps/ is a possible initial cluster (though it is rare, and pretty well confined to words of Greek origin) but /pt/, /pn/ and /kt/ are not possible initially, and get simplified (as does /kn/, which was possible in Old English).
All these clusters are normal medially, and generally split the consonants between syllables. Thus uptake and archaeopteryx contain /pt/ medially.
