If you've found a BSAC club that you like, then I would recommend joining. A BSAC certification would be recognized/accepted world-wide. In practical terms, that means you would be able to present your certification card to a dive centre and be allowed to book diving. Because of the club/mentoring system, coupled with the 'cold-water' diving mindset, BSAC is recognized as providing an above-average quality of diver training.
PADI is the largest diving training agency, providing the majority of global diving certifications annually.. The majority of diving shops/operations in most regions are PADI. This would make PADI the most recognized agency - in the sense of 'brand awareness'. Some PADI dive centres do offer a club membership (this is increasingly true in the UK), which would also bring simular mentoring/club benefits as BSAC.
The essense of an entry-level training course is very simular with either organization. A diver has to learn X, Y and Z skills, whichever agency they choose. However, PADI applies a course structure which is globally relevant, which tends to make the OW course into the lowest common denominator. Because of this, it relies on continuing education courses to supply important regionally specific training (for the UK, that might be a separate drysuit course, extra training for DSMB deployment, a separate course to calculate tides and water movement etc etc). In contrast, BSAC tends to be more inclusive with these skills - as it's focus is to train divers for UK diving.
One potential downside to BSAC training is that courses can be slow. This is actually a benefit in terms of progressive skill development, but can cause issues if you were seeking to complete training before a set date (say, for instance if you were going on a diving holiday). BSAC relies upon volunteer instructors giving their time to conduct training, so it is often not possible for clubs to schedule training to suit individual member preferences. Courses can take many months to complete. In contrast, PADI dive centres are commercial - so they will exhibit more flexibility and motivation to 'get your business'. Most PADI operations will offer certification based on 5-7 days of training; whether that is completed one-night-per-week (like a BSAC club), over several weekends, or in a non-stop 1 week course.
Another potential downside to BSAC training is that it is very much determined by the attitude, mindset and motivation of the individual club that you join. There are some great BSAC clubs in the UK, but there are also some very clique and/or backwards minded clubs. If you like your local club and get a good 'vibe' from its members, then you probably have nothing to worry about.
The same is true of BSAC/PADI instructors. The overall quality of training is always determined by the individual merits of the instructor who provides it. I know some amazing PADI instructors....and many terrible ones. The same is true of BSAC. If I had to generalize though, I would say that BSAC offered a higher consistency of quality instructors, because it is a smaller organization and it doesn't 'push' people towards instructor level for commercial reasons (PADI makes lots of money from their instructor development program). That doesn't mean you can 'write off' PADI training though - you just have to be selective about which instructors/operations you use (but isn't that true of all things?).
I, personally, don't subscribe to the 'cold water' training must be better mentality. You will encounter this, I am sure. There are pros and cons to diving in any water environment and what matters is the quality of training, the quality of the instructor, and the divers individual mindset; in how they prepare for those conditions. From my personal experience, the divers who shouted loudest about the superiority of their training and experience (whether that is cold-water, or caves or whatever) tend to do so because of other, more personal and egotistical issues.
Also, be aware that agency qualifications are inter-changeable. You could do a PADI open-water certification and then join a BSAC club for further development. Vice-versa, you could do your training with a BSAC club, but still take subsequent qualifications with PADI (or other agencies).
In respect of long-term development, I think that BSAC offers more. Both agencies are roughly simular at Open-Water to Rescue Diver level. Beyond rescue diver, PADI only offer the 'Master Scuba Diver' certification (which is nothing more than a 'plastic clap on the back' for doing more specialist training courses) or a Divemaster qualification (which is a professional qualification to supervise divers). In contrast, BSAC offer the Dive Leader, Advanced Diver and First Class Diver ratings, which permit development far beyond the skill-set which the PADI system provides.
