I own an Uzi and have shot most of the HK subgun variants. Big difference that nobody has mentioned is cost.
Full Auto Uzi or Uzi Mini = $7k to $8k
Semi-Auto Uzi clone = $800.00
Full Auto HK MP5 = $17k to $20k
Semi-Auto HK clone = $1,200.00 to $2,200.00
If someone were to just give me whichever I wanted for free, I would choose the HK. If have to pay, hands down, the Uzi a superior value. The Uzi aftermarket is huge, lots of parts and accessories available everywhere, HK parts and accessories, much more expensive and harder to find.
Both are great guns, overall, the HK is more sophisticated but I think the Uzi is better for most people, short of professionals who are in combat.
As far as accidental discharge, I own the semi-auto Uzi, which fires from the closed bolt, the gun is quite safe. You could research the basic variants of the model A, model B and any issues with accidental discharges at
www.uzitalk.com If you practice safe gun handling, you will not have any issues, Uzis are pretty popular in subgun competitions and are great guns.