D4v1d-Gr43b3r's excellent review talks about the functionality of Physarum, so I'll stick to its impact. Put simply, Physarum is the best single card answer to nearly every non-barrier ICE in the game. You'd think this would make it a strong contender in Criminal lists, but no, it's awkward and mostly a worse Boomerang because you can't blindly put this onto ICE for fear of it being a Barrier and getting blanked or getting purged out.
Of course, it's not Criminals that are playing Physarum, it's Shapers. At 3 influence, it does strain deckbuilding slightly, but you only need one copy to tutor/recur with cards like Self-modifying Code, Simulchip or Muse at potentially instant speed, meaning you can avoid its downside of potentially putting it down blind and discovering it was a barrier all along. If you're Arissana Rocha Nahu: Street Artist, it's a Trojan you can slap on from hand too and break any non barrier for (usually) 2-3c. The fact that it bypasses on when encounter begins means, as D4v1d-Gr43b3r points out, that it skips many of the "on encounter" triggers that lots of taxing ICE such as Tollbooth rely on. Basically, if a Shaper has SMC down and ~5c (or a Simulchip with it in heap), they can face check nearly anything without consequence (the only favourable interaction for the Corp is Anansi, as seen by the ruling above this page).
The effects of this on Corp deckbuilding, just due to this card's existence, are extremely severe. Since Shapers have been the clear dominant force of the RWR meta and Physarum is a common one of in almost every one (except Rielle “Kit” Peddler: Transhuman), playing strong face check Sentries or punishing Code Gates is strongly discouraged (not entirely out, Jinteki and HB have enough scary face checks that the one Physarum can't hit them all with cards like Saisentan or Gatekeeper) as a Physarum Entangler will break them for pennies. Large and expensive Sentries/Code Gates like Cloud Eater, despite being strong from a strength/subs/facecheck point of view are completely unviable when "rez for 10, break for 3" is the standing. I won't deny the weakness to Mavirus; if the Corp has an upgrade on the table it can be risky to use a Physarum to break, and currently we do see Mavirus in Rush PD and Ob Superheavy Logistics: Extract. Export. Excel. decks).
Ultimately, though, Physarum Entangler is one of the strongest Shaper cards to ever be printed in Criminal. Sadly, it's not even the only Criminal card Shapers found more reliable use for (Cupellation). As you can probably tell, I really don't like this card; it lacks so many of the downsides of other ICE cheating methods NSG has printed (Botulus is slow, Boomerang and Slap Vandal lose to certain ICE and S-Dobrado and Inside Job can be played around via positionality and need to be hard drawn) by being an immediate and fast solution to too many ICE in the game, and has only further strengthened the "ICE doesn't matter" endgame lots of Runners end up in. I would not be against seeing the card go on the banlist, just because the strain it puts on Corps and efficiency at which it cheats ICE on encounter effects compared to something like Femme Fatale is absurd to me.
Update: It's fine as is. Well balanced!
— DDDydra