Yes, they both have the "broken" meaning, with subtle differences. Unfortunately there are not many situations where you can simply replace one with the other - everyone will understand but you will still sound awkward if you use the wrong character. Plus there may be regional differences!
I suppose if you had to generalise, 破 is used when the breakage is obvious, there's a clear mark or split or wound, whereas 坏 is for things that are broken in more subtle ways, or less obvious. 坏 also tends to be used by default, when the situation is ambiguous (other characters for "broken" include 碎, 败, 漏, 裂 etc.)
So here's where you'd use 坏 and why:
手机坏了 (broken mobile phone), because the phone may look fine
鱼坏了 (the fish has spoiled), because the fish may otherwise look fine
Here's where you'd use 破 and why:
纸破了 (the paper is broken), because the paper has a visible hole or rip
鞋破了 (the shoe is worn), usually because you wear through the sole so that there's a hole
碗打破了 (the bowl is shattered), because the bowl is now in many pieces
Also watch out for words that use 坏 or 破 since, as you may know already, Chinese words cannot be mixed-and-matched by replacing similar characters in them. E.g. you cannot replace the "破" in the word "破裂" with "坏" to get "坏裂", as Chinese doesn't work that way.
For a bit of etymology which somewhat explains the subtle differences, 坏 has the "earth" radical and originally meant "unfired clay", so outward appearance is fine but it's not really "made"; 破 has the "stone" radical and originally meant stones shattering, so there's a visible, obvious breakage.