Before WWII, the Innocenti factory at Milan produced steel scaffolding, water pipes, and conduits for the building trade; after war broke out, the factory was converted to produce cannon barrels and ammunition. It is no surprise, then, that Lambretta scooters feature a steel tubular frame, unlike the Vespa's load-bearing monocoque body. Although Vespa snobs deem this heretical, the Lambretta's tubular frame does allow the scooter to be chopped, just like a Jesse James or Orange County Choppers creation on a much smaller scale. Additionally, the Lambretta scooter features a centrally positioned engine, unlike the Vespa's engine block which is placed as far away from the driver as possible, next to the rear wheel.
The steel tubular frame and centrally-mounted engine provide enough grounds for Vespa purists to hail the Lammy as not a scooter, but a motorcycle in sheep's clothing! Indeed, in 1951 the Motorcycle Manufacturer's Association (ANCMA) set up a meeting with Piaggio, Innocenti, and several other major Italian scooter manufacturers to reach an agreement as to what traits a scooter possessed. According to this meeting, a scooter is "an engine-powered vehicle without pedals for human propulsion characterized by an open frame in the front and rear parts joined by a continuous footrest, and by two wheels the diameter of which is no greater than 12 inches."
In terms of sales, Piaggio had nailed the Italian market by 1950, when the first Lammy rolled off the Lambrate production lines, so Innocenti concentrated heavily on overseas. In the late 50's and early 60's, well over half of Innocenti's factory production was destined for overseas, while Piaggo's exports lingered around 40%.
As mentioned above, Lambretta scooters are typically sportier than their Vespa counterparts. For instance, the 1962 Lammy 175cc Sport was the first scooter to be outfitted with disc brakes. In 1965 Marlene Parker ran her highly-modified 200cc Lambretta to speeds reaching 130mph, although the time was deemed an "unofficial time." However, stock Lambrettas were often not the most efficient machines, and although they produced more power than Vespas, the two makes were often a disappointing draw on the race track.
Just as a quick tangent, it's interesting to note that both Lambretta and Vespa scooters were designed by aeronautical engineers, Pier Luigi Torre designing the former and Corradino D'Ascanio designing the latter.
Dregni & Dregni, The Illustrated Motorscooter Buyer's Guide
Elissa Stein, Vespa: Style in Motion