Hmm... Not really sure what to say to this. I've tried all of the classes and I have most of the races now (still missing salarian infiltrator and drell vanguard, if I remember correctly), if not very many new weapons worth mentioning. Speaking on the pure basis of the classes, however, they all have their own unique specialties, that I've found.
Engineers are practically invaluable at some times, especially due to their abilities that distract or otherwise provide a secondary point of attack on tougher enemies. My quarian engineer (lvl 19 atm) has soloed both Brutes and Banshees on silver difficulty, and I have managed one brute on gold by myself while the team was otherwise occupied.
On a different note, the vanguard excels at dealing exceeding amounts of damage to single enemies, or small groups, especially the humans with their nova and shockwave for CC, and the Asari, with her aoe strong melee and her stasis (completely invaluable with phantoms, makes them so much easier to kill).
The adepts, while physically weak, especially the drell, can deal large amounts of damage to any single targets with their range of biotic abilities. Both warp and reave are excellent at depleting enemy armor.
The soldiers, I've found, function as a very effective powerhouse for simply spamming weapons fire. If other classes can keep the enemies from reaching too close to them, from medium to long range, the soldier can be exceedingly powerful, especially with their various explosive accoutrements. The Krogan soldier on bronze is a tank, and he can mow through enemies with a proper shotgun/assault rifle. On silver, he can still be just as effective, but he has to be much more conscious of his surroundings, lest he be overwhelmed by cannibals/marauders shooting from the side/behind.
Sentinels were actually the second class I promoted, the first being vanguard, and I played most of my time as a turian. I found, playing as other races as well, that the sentinels fit in well as more of a Jack-of-all trades. The turian, for example, has overload, for shields and barriers, and warp, for barriers and armor, making them particularly useful against enemies such as banshees, provided they stay clear of that instant-kill range. The krogan with tech armor becomes a veritable sponge for damage, provided, as with the soldier, that they manage to steer clear of choke-points and rushing away from the rest of their team. whether they focus more on their grenades or their incinerate, each option can provide a different style of aid for the rest of the team.
Infiltrators are actually class I've played as the least (despite the Widow being my first spectre pack unlock.), so I can't offer much besides the obvious. They provide powerful support, as long as they remain at range, and their tactical cloak is immensely valuable for mission objectives/reviving downed teammates without being seen. That being said, at the times where I've been revived by a cloaked infiltrator, I was generally so enveloped by enemies that no sooner was I able to regain my feet that I was knocked down again because of the sheer number of surrounding hostiles.
