Hey there - I'm actually writing a Grad thesis in part on slave soldiers in the Islamic world - this is a very very interesting question.

To start, Mamluk is a pretty broad category while Janissary is very specific. Mamluk is more or less interchangeable with the word ghulam (pl. 'ghilman') and describes the slaves (of primarily nomad Turkic origin) who were brought in to serve as a standing military force in the major courts of the Islamic World after the Abbasid revolution shifted the military focus away from traditional Arab tribal organization towards more centralized modes of power. While Janissaries are clearly made with this practice in mind, they have their own distinct identity owing to their context.

Key things in common:

• Esteem in Society - Although slaves, both groups occupied a high place in their respective societies. As the backbone of their dynasties' military power (especially given the relative unreliability of levies and 'feudal' troops like Ottoman Sipahi - who often proved undisciplined and varied widely in quality) it was crucial to the Sultan (or Caliph, Emir, what have you) that these warriors were loyal and content. If the relationship between the sovereign and his slave soldiers was strong, they served as a deterrent against conspiracy from within as well as against attack from outside - and their more professional nature made campaigns of expansion much more viable. Levies and 'nobles' (such as the sipahi) needed to return home at some point to manage their households (or be artisans/farm for the lower classes) but a mamluk or janissary's sole responsibility was to fight for the Sultan.

• Collective Identity/Caste - Both types of slave soldiery had powerful esprites de corps and strong cohesion among themselves as a caste. It is for this very reason that 'mamluk' dynasties such as the eponymous ones in Delhi, Iraq, & Egypt rose to power as well as the Ghaznavids and Khwarazmians in Khorasan/Persia. Loyalty to the crown and loyalty to the brotherhood were - in practice - very close to equal in the eyes of many of these slave soldiers. This often resulted in a strong aversion to reform - which at times led to fearsome & renowned slave soldier forces (including even the Janissaries later on by the early modern era) to become increasingly obsolete in the face of new strategies and paradigms.

• Meddling in Politics - For both the Ottomans and their predecessors, the slave corps served a praetorian role in every sense of the word - including meddling in dynastic affairs and acting as kingmakers. Without the support of the slave-soldiers, one could never hope to be Sultan. There's even an interesting practice among the Janissaries where they would protest a Sultan they disagreed with by refusing to partake in their daily meal 'from his table' - symbolically rejecting his legitimacy. Despite being slaves, they were just as involved in politics as anyone else in the court.

• Part of the Sovereign's Household - As hinted at above, Janissaries and Mamluks were paid directly by the Sultan (fitting given that they were his property) so as to minimize any chance of conflicted loyalties. Ideally, this rendered a strong Sultan much less beholden to his governors and to local elites. Local rulers knew as a fact that to rise up would be to bring the slaves of the Sultan down upon them while they might very well be able to muster little more than local militia or mercenary forces. Indeed from its very inception the practice of having a ghulam corps is intimately linked to curbing the power of noblemen or local elites.

Key Differences

• Recruitment -
  Whereas Mamluks and other older forms of Ghilman soldiery were drawn from slave markets on the fringes of the Islamic world (chiefly Khorasan, but the Balkans, Circassia, Africa, and India were also fair game) and were by a large margin embedded in old Turkic military traditions, the Janissaries were recruited by way of a systematized bureaucratic levee known as the Devshirme. The Devshirme consisted of- a tribute of christian boys in Anatolia & the Balkans at or below the age of 18 collected as a tax. These Christian boys were then brought back to the capitol to be trained in warfare & converted to Islam. Curiously, there was a tradition barring a single family from having multiple sons taken per Devshirme cycle but even so many families offered up more sons in the hopes that they could achieve great things through the Devshirme. (It's also important to note that slaves from the Devshirme would not necessarily end up as Janissaries - those more suited to policy, theology, or scholarship were sorted to serve as court slaves & bureaucrats rather than as soldiers.)

• Military Focus - Since the Mamluks were steeped in the martial culture of the nomadic Turks, the cream of their military corps were cavalry based - heavily armored & skilled in bow, lance, sword, and mace. By contrast, the Ottomans (being Turks themselves) started out with a strong cavalry force in the form of the Sipahi and thus developed the Janissaries as an infantry force both to expand their combined-arms approach to warfare and to create an effective counter to the power of the cavalry-based local nobility. Turks & other Muslims were specifically barred from the Devshirme for centuries in order to maintain the separation of power. In addition, Janissaries were barred from marriage and producing heirs until the reforms of Selim II in 1566.

• Religious Zeal & Sufism - Another important note of contrast was that the Janissaries were far more steeped in traditions of sufism and mysticism than their earlier counterparts (Fun fact - the janissary headdress actually represents the sleeve of a sufi mystic who blessed the corps on its founding). The Sultan was established both as their physical master/owner and as their spiritual guide (very much like the Qizilbash of Safavid Persia), though in later phases of the empire this dynamic waned.

• Revolt vs. Usurping - Although as I said before the Janissaries meddled frequently and irritatingly in Ottoman politics, they never seized power directly for themselves. It even took until Sultan Osman II's attempt to disband the corps in 1622 for them to even threaten a Sultan's life - and according to Ottoman accounts it was their killing of Osman which eventually lead to their disbandment in the 'Auspicious Incident' of 1826. In the end Sultan Mahmud II was forced to use a seperate modernized army he had raised in secret (the Sekban-ı Cedit) in conjunction with the Sipahi (whom had long despised the janissaries) to put down the ensuing revolt of over 100,000 Janissaries.

By contrast, numerous mamluks had usurped the throne for themselves when their masters became weak - some founding dynasties (especially like in Egypt) that lasted for centuries.

I hope that helps some. If you have more specific questions or sources I'd be happy to weigh in or share. Also - as a fun parting note - the Mamluks fought the Janissaries numerous times, both during the Ottoman conquest of Egypt and their subsequent revolts against the Sublime Porte :D

Edit: formatting