Considering this is a PS thread.. it's not surprising that more people voted for the PS1!
That said, I own both bags and I think you need to break down further HOW you intend to use this bag ie. are you planning to use it more as a crossbody, handheld or shoulder..
From my personal experiences - and please bear with me over its length! - I think the Bal Velo is perfect as a crossbody and for all the things you mentioned -
everyday shopping, sightseeing, coffe-ing, travelling/going on tours, etc. In fact, it's pretty much what I've been using my Bal for.
Why it's working for me as a 'lazy bag' (I love this description!) is because it's very slouchy and just rests against my hip like second skin - so it's perfect to wade through crowds and public transport. The second reason for the Bal being the better crossbody is because of its lengthier strap drop as compared to the PS1. Being 5'7", the drop of my PS1 large's strap is several inches too short to rest comfortably at my hip.
However, if you're more likely to carry it shoulder and use it for more formal/dressy occasions, I'd say go for the PS1. Firstly, it's much more structured and tends to keep its shape when carried. I find the Bal less ideal as a shoulder - it's a wee bit too long when I wear the strap, unless you are very tall. Also it's quite uncomfortable (almost impossible imo) when I wear the handles. But this shouldn't be a problem if you have skinny arms.. unfortunately I don't! So no shoulder for me...
Handheld-wise, I don't think there are significant differences - both have pros and cons. The pro for both is that they look great handheld!!!
The con for the PS1 is that it's quite difficult to close once it's open - the front 'splats' out so you're left standing/balancing on one foot to get the clasp close again. Though it could also be I'm just clumsy by nature!
As for the Velo, it's pretty deep so it can be quite a bit of a dig to get anything from your bag!
The bottom line is I think both bags are great!! If you already own a PS, I think the Bal would make a lovely addition. But if you don't already own a PS.. well.. I guess only you can decide which bag to go for!