I don't think it is correct to attribute these differences to opiates vs. opioids. Some opiates are associated with some of the effects you are describing.
The different effects are caused by:
1) Differences in binding to mu subtypes. Mu is generally thought to mediate euphoria, analgesia, and respiratory depression. There may be mu subtypes produced by alternative splicing, and some compounds may be selective for only one subtype. I've heard of mu-1, mu-2, and even mu-3 sites. There is also evidence for a mu subtype that binds the morphine metabolite morphine-6-gluconuride but does not bind morphine.
2) Selectivity for different opiate receptors. There are mu, delta, and kappa receptors. Opiates tend to bind to all three with preference to mu, but selectivity can vary widely from compound to compound. Opioids are often much more selective for one of the receptors. Fentanyl, for example, tends to selectively activate mu. The three receptors produce different subjective effects and so ligand selectivity can markedly alter the profile of effects.
3) Off-target effects. Opioids and opiates can produce effects by blocking NMDA receptors (methadone, levorphanol), the dopamine transporter (meperidine, tilidine, levorphanol), the setotonin transporter (meperidine?), and glycine receptors (thebaine).
4) Effects of metabolites. Some opioids yield metabolites that have different actions than the parent drug. Normeperidine produces convulsions and is not an analgesic.
Opioid receptors can also form dimers and it is possible that ligands that bind to the dimers preferentially may have different effects than ligands that bind to the protomers (individual receptors).