Strong Uniform Total Order vs Weak Uniform Total Order The definition for SUTO and WUTO is the following:

Suto


  If some process TODelivers some message m before message mʹ, then a process TODelivers mʹonly after it has TODelivered m.


And for WUTO


  If process p and process q both TODdeliver messages m and mʹ, then p TODelivers m before mʹ if and only if q TODdelivers m before mʹ. 


I have a hard time getting my head around this. For me, both of them sounds the same...

Any help?
Consider for example two processes $p_1$ and $p_2$, with $p_1$ that delivers in this order $m$ and then $m'$.
In SUTO, $p_2$ will either deliver $m-m'$ as $p_1$ did, or it won't deliver $m'$ at all.
But why?
Because if $p_2$ will deliver, for example, $m - m'' - m'$, then every process can deliver $m'$ only after $m''$.
This means that once $p_1$ and $p_2$ don't deliver the same messages, the  following  messages delivered by $p_1$ and $p_2$ must be disjoint.
In WUTO instead it's possible to have $p_1$ that delivers $m - m'$ and $p_2$ that delivers $m - m'' - m'$.
This example is taken from Baldoni, Cimmino and Marchetti, A classification of total order specifications and its application to fixed sequencer-based implementations.