It is popular for various persons to hypothesize that Christianity was some sort of outgrowth of the Essene movement in Judaism, and from there to hypothesize until Doomsday in an effort to make Christianity a natural religion. They do not need the Essenes to try that course, naturally, but since it is a popular way of business, we have decided to offer a brief outline of differences (and similarities).
Our sources are Geza Vermes' The Dead Sea Scrolls: Qumran in Perspective, Schubert's The Dead Sea Community [Sch], and VanderKam's essay "The Dead Sea Scrolls and Christianity" appearing in Understanding the Dead Sea Scrolls [Kam].
Our first point is somewhat of a surprise, for Vermes  tells us that it was two scholars (Teicher and Baer) who first suggested a connection between Essenes and Christians. The removal of such a thesis to the province of "fringe" scholars like Allegro, Thiering, and Dupont-Sommer, and thereafter conspiracy theorists, is therefore one that has been developed after consideration by scholarship -- not because it has never been tried or suggested before.
It is well to acknowledge that one will certainly find much that is similar between these two groups. The error of those who find derivation is what we might expect: similarities are highlighted, while differences are ignored, and the similarities are found in places like universal traits of religious traditions, OR universals within the pale of Judaism (i.e., a parallel would also hold to the Pharisees, and thus be pointless in context).
In the end however, Vermes concludes that "the two ideologies differ fundamentally"  such that a "common roots" explantion for both movements is to be preferred, though without adding that there would have been NO influence at all. To that end, here is a simple list of fundamental differences:
- There is "heavy emphasis on the punctilious observance of the Mosaic Law" at Qumran, versus the "peripheral importance" in the Gospels.
This alone, Vermes tells us, makes a linear descent seem "extremely improbable".  Schubert [144Sch] notes for example that though Jesus allowed life-saving rescue on the Sabbath, the Essenes forbade it "if any instrument was required" to do it . Essenes were more fanatical about the Sabbath than the Pharisees Jesus rebuked.
- The Qumranites saw their movement as the fulfillment of OT prophecies. Christianity saw OT prophecy fulfilled in Jesus and activities related to him.
For example whereas Essenes applied Is. 40:3 to themselves, Christians applied it to John the Baptist. John himself, sometimes regarded as an Essene or a rogue Essene on the basis of his location, may well have had associations with Essenes, but Schubert  retorts that it would be odd if such an association were not reported by Josephus (as well as the NT) and that there was another such "desert hermit," one Banus reported in Josephus, who lived in the desert and even baptized himself daily without being an Essene.
- The Qumranites expected a restoration of Temple worship. Christians expected the Temple to be destroyed, to be replaced by God and Jesus [215; cf. Rev. 21:22] and a Jerusalem with no Temple.
Given the Temple's central role in Judaism, this difference is itself monumental.
- Celibacy was compulsory for most Essenes, but only a limited ideal for some in the Christian movement . For Essenes celibacy was a matter of ritual purity, but for Jesus, a goal for those who would serve in the Kingdom wholeheartedly.
- The Essenes would not engage in controversy with outsiders . In contrast Christianity was a missionary faith in constant conversation with outsiders.
- Related to this, Christianity welcomed sinners and the unrighteous to repent while at Qumran there "rested elements of intolerance, rigidity and exclusiveness."  Christ purified the sinner; the Essenes avoided the sinner.
- Jesus taught live for enemies. The Qumranites taught hatred for enemies , which was tied in to their view of the vengeance of God on their enemies.
- Christians baptized once and for all; the Qumranites took repeated lustrations for purity. [Sch, 129]
- The Essenes apparently believed in two Messiahs, one priestly, the other Davidic [Sch130]. In Christianity Jesus was seen as filling both roles (cf. Hebrews).
A worthy side note on this topic is that the Essenes expected messianic salvation to be revealed in the wilderness -- in direct opposition to Jesus' warning to not follow anyone who claims that that is where the Messiah will be found [Sch134; cf. Matt. 24:46). The only clear role the Essenes assigned to their Messiahs was to preside at the communal meal [Kam, 196]. There is no indication of atonement for sins.
- Jesus repudiates any idea that his followers will conduct a physical war to begin the Kingdom, whereas Qumranites saw a war as the instigation, with angels fighting on their side [Sch136].
- An oft-noted similarity is the sharing of goods as communal property (Acts 2:44-5).
This is offset by two points: 1) This would not have been atypical for any social "ingroup" of the period to some extent; 2) at Qumran, initiates were not allowed to share in community property for a year, and then their property was handed over to the Bursar who held it for another year. Only after that was the property merged with that of others [Kam, 193].
- Another claimed similarity is that of a communal meal with bread and wine. Aside from that these ancient staples were often used for communal meals in many groups, here again there were probationary periods before which the initiate could eat (1 year) and drink (2 years) [Kam, 194].
- Paul and the Qumranites both made use of Habakkuk 2:4, "The righteous live by faith." But whereas Paul uses it to argue that faith, not works, is the way to salvation, Qumran's Habakkuk Commentary uses it to encourage fidelity to the law! [Kam, 199]
Against this, we have similarities of the sort best explained by a common root in Judaism (i.e., appointing groups of 12 to serve, copying the 12 tribes; eschatological expectations; administration; dualism) or in the political and social situation of the day (i.e., rejection of the corrupt Temple apparatus by some members; sharing of goods among an ingroup).
Vermes does see Paul adopting Qumranic symbolism for his own purposes  and also suggests that Christianity borrowed certain Essene models for "organization and religious practice." But this was done, he adds, after the manner of a nascent movement looking for practical tips to survive and operate.
A couple of final notes are in order:
- Any critic seeking to make light of the lack of mention of the Essenes in the NT as a proof of identity should know that the Essenes are also conspicuously not mentioned in the rabbinic literature . Schubert [Sch123] adds that it is uncertain what the Essenes even called themselves; "Essenes" as a term is not even found in the Qumran texts.
Schubert suggests Essenes may have been included en bloc as among those who were "looking for the Kingdom of God" along with righteous persons like Joseph of Arimathea.  Some have suggested as well that the male water carrier in the Gospels was an Essene. Some suggest that Luke 16:8-9 refers to the Essenes (who called themselves "children of light") but not all agree to this.
- Having seen such differences as above, the resort will inevitably be to, "Well of course, Christianity came from Essenism, but evolved into a different movement." At the point this is used, the data is being molded to fit the theory rather than the other way around. The differences above are so radical that "evolution" would be a far more complex theory than independence and would require a great deal more tangible evidence than, "John the Baptist hung around in the same neck of the woods".
The differences are enormous and cut to the fundamentals of each movement, so that foundational independence, with common roots in Judaism and influence only of the sort found among any pair of social movements in the same political and religious milieu, is a far simpler and easier explanation.