Can someone explain me which one is correct and why?
The first one seems to be more logic to the last one. But both of them are strange. I didn't get the point that what would you like to say.
To explain at least, the last sentence lacks object of verb "speak".
If they are declarative sentence, it's better to say 我对我的妈妈说了很多话 or 我对我的妈妈说了很多. Both of these are reasonable, but don't mean the same. The 2nd one seems to pull other points more, but 1st one didn't have this meaning. It's complicated, wish that I've make it clearer to you.
Depending on the context.
說很多話 can mean that you habitually talk to your mother a lot.
說(了)很多 means you have said a lot of things (regarding a specific subject).
First, in Chinese if you don't want to emphasize "我的，你的，他的"，we usually remove "我的”，我对我妈说....or 我对妈妈说 is more natural.
Secondly, it should be added a complement "了" after the verb "说" to indicate the action is finished. So it could be 我对妈妈说了很多话。or 我对妈妈说了很多。
Finally, 说了很多话 means that you speak a lot "sentence" to your mother, while 说了很多 means you talk a lot topics or opions with your mother.
actually，we oftensays like this, “我和妈妈经常聊天”，or，“我对妈妈讲了很多”… we usually drop the term “我的” if it is implicit but clear and unambiguous.
in most cases they are the same. Chinese has omissions of objective, that is, many verb has the most frequently used objective, like 说(话) and 吃(饭), so if you just use 说/吃 without objective, the objective could be regarded as 话/饭 by default.