当要表达“在某人的帮助下”时用with one's help.当要表达借助某人/某物时用with the help of sb/sth.
With one's help意思是“在某人的帮助下”,其中的one's是形容词性物主代词或名词所有格.
With the help of则是加名词性物主代语或名词.
with the help of sb. == with one’s help 在某人的帮助下
其实我们可以看到例句都是可以互为改写的.
with one's help例句

With the help of a passer-by, I carried the injured person into a shop.
在一个过路人的帮助下,我把伤者抬进了一家商店.
// With a passer-by's help, I carried the injured person into a shop