Physical benefits of slow and deep will be slightly different to faster styles like vinyasa in that your heart rate won't be as high but you'll become more flexible and if you're doing say a strong but slow hatha class then you'll find your muscles do start burning at times!
Slow styles that might be interesting (not a comprehensive list!):
Yin- this is all sitting and lying down and great if you have injuries. Props are used so postures can be held for 1-5 minutes. The idea is that muscles start relaxing and stretching after 30 seconds or so, ligaments and tendons take 2-3 minutes so these tissues get stretched and become more flexible although this takes a few months to achieve. Restorative yoga is very similar.
Scaravelli- a very precise form of yoga from the sadly now deceased Vanda Scaravelli (have a look at some of the stuff she was still doing at 80+!). She adapted classical postures to focus more on alignment and using very specific muscles while relaxing others. It takes a fair amount of concentration so is really good for destressing as you'll be mostly thinking how on earth do I relax my bum... Classes tend to build up to a peak posture.
Strong and slow hatha- this is more about finding a teacher who covers the classical postures but perhaps builds up to a peak posture or holds postures for a while and works very precisely rather than rushing through them (as I have found a lot of teachers do). Be prepared for muscles to burn in standing postures but you will definitely build strength.
Kundalini- not a style I know much about but the focus seems to be on breathing rather than postures. It doesn't seem to go through the classical poses like other styles.
Sivananda- slow practice with only a few postures compared to other styles.
I practice Ashtanga the most regularly and find it very meditative but it is a faster, more sweaty form of yoga and I only started after several years of hatha. You might find as you get more used to the postures that you want a different style of practice, that's totally normal and most people I know have been through several styles on the way to their current preferred option. It's important to like your teacher and class but not be too beholden to them and be prepared to move on at some point in time. A good teacher will not mind at all. Be sceptical of classes where the teacher is up at the front demonstrating the entire time rather than coming round to check the students and making adjustments, I've been to too many classes where I look round and see people doing terrible things where they might injure themselves and the teacher doesn't notice.