The advantages of bioengineered weapons are autonomy and self-replication. Truly advanced bioengineering would allow for the creation of terrifyingly effective weapons. This answer will cover some various archetypes of potential living weapons.
The simplest biological weapon to produce is one that humans are already working on. What we think of as “biological warfare”. This equates essentially to creating potent diseases. If your aliens use DNA and RNA then their engineering technology will likely work on our existing pathogens. The aliens will have to do a lot of analysis to figure out how earth biology works, but once they have the basics they will be able to create diseases more devastating than any humanity has ever faced.
Viruses tailored to wipe out entire human populations in days are only the beginning. Bacteria, fungi, and plants designed to degrade infrastructure and electronics, destroy crops and livestock are also possibilities. Entire ecosystems can be wiped out by the extinction of vital keystone species. Your alien genetic engineers could create blights to poison our soils, our seas, and our air. If they so wished they could destroy not just humanity, but life as we know it.
These are truly weapons of mass destruction. Nuclear or even an antimatter weapons merely render a single planet uninhabitable. These are weapons of genocide, contagious and persistent enough that refugees will bring with them to other worlds repeating the process over and over again. A single space-born spore would be capable of destroying an interstellar civilization.
As far as conventional warfare goes I think it’s important to point out that there’s no reason it would be impossible to use biotech to manufacture weapons made out of metal and plastic and powered by electricity. I imagine you want your weapons to be squishy and gooey and squirming though, because otherwise what’s the point?
With that said, a lot of what is possible for your aliens in terms of weapons is going to depend on just how advanced they are. Our current metal, plastic, and silicon based technology is vastly superior to anything alive on earth and certainly anything Earth is capable of evolving. But your aliens are presumably capable of spaceflight using living spaceships so clearly they have improved on mother nature. Organic materials like spider silk can be quite strong and even surpass artificial materials in some areas. Fortunately for the aliens, the strongest material ever tested, graphene, is actually carbon based. In theory your biotech could be capable of synthesizing various allotropes of carbon. Weapon barrels, ship hulls, bunkers, and armor plates could be composed of monomolecular sheets of graphene, grown carbon by carbon by biological processes, sandwiched between proteinaceous layers that provide additional mechanical strength as well as thermal and electrical insulation. This sort of arrangement would likely be incredibly effective as a form of armor. On the offensive front biological processes are quite capable of producing potent poisons, explosive chemicals, strong acids, and the aforementioned diseases. This means in terms of conventional weapons you can easily imagine a big biological tank that your aliens can get in and shoot gouts of flame or lob explosive rounds from. But let’s try thinking outside the box. Why require your tank to be piloted at all? It’s a living thing, let’s make it intelligent. This is where the advantage of autonomy comes in.
Imagine a fruit fly. A tiny, 2.5 mm long, flying insect. It has 250,000 neurons that allow it to walk and fly, find food, mate, lay eggs, and avoid our clumsy attempts to squish it. It’s easy to imagine how a brain of that size could be programmed to walk and fly, find humans, deliver diseases, poisons, acids, and explosives, and evade basic countermeasures. I use fruit flies as an example simply to indicate just how small a brain is needed to enable fairly sophisticated behaviors that will make for effective weapons. It is this ability to produce autonomous, intelligent weapons that is the greatest strength of organic technology in “conventional” warfare. Your missiles won’t be fooled by flares or other countermeasures. Your tanks will be savants in calculating trajectories. Your landmines will distinguish between friend and foe.
By autonomy I’m not just referring to the ability of these weapons to function without their masters, but also their independence from any logistical structure. They are fully combat effective without any outside input in the form of supplies or commands. A surface-to-air missile is expensive. Humans have to mine and refine all sorts of materials, shape them through several successive steps, create incredibly advanced microchips, assemble it all together, transport it out to the front lines, and put it on a launcher. Imagine if you could simply plant a seed on the end of a nutrient hose and grow an intelligent missile. As it grows it chemically refines its nutrients into an explosive warhead and fills an internal cavity with fuel. It creates its own launching structure and grows aerodynamic, adjustable fins and a gimbaled rocket nozzle. At its nose it forms an advanced eye constantly scanning the sky and just behind it a brain instinctively programmed to destroy enemy craft.
While this missile may be significantly less effective than a human-crafted one, its important advantage is it is free. Made organically from the water, soil, and sunlight of a world it didn’t have to be transported to, it is simply grown right where it is needed.
Now imagine that nutrient hose which provided everything the missile needed also grew from a seed. A seed that was planted in plenty of sunlight on the banks of a river that grew huge arrays of photosynthesizing leaves for energy. The hub sends roots deep into the earth and water to harvest the necessary materials and uses them to feed a whole battery of missiles, interceptors, ground forces, and of course, seeds for more hubs. An autonomous, intelligent, self-reproducing weapon that requires no logistical support.
Your biological weapons might not be as fast, or as sturdy, or as powerful as their man-made counterparts. But they are going to be numerous. They are going to be smart.