啦 and 嘛: sentence-final particles
【六23】语气助词 / particles
Tone colorsā tone 1á tone 2ǎ tone 3à tone 4a neutral
啦 is 了 + 啊 fused: it closes the sentence on a cheerful, warm or exclamatory note, to announce a change (车修好啦! "it's done, the car's fixed!") or to hurry someone along kindly (好啦好啦 "all right, all right"). 嘛 says "it's obvious" or "come on": it goes on a point you are reminding someone of (我早就说过嘛), on an invitation you want to make insistent (你说嘛), or after a topic to mean "after all" (小孩子嘛).
Both give speech its colour and never ask a question: to ask, keep 吗 or 呢 (level 2), and 了 for a change (level 1). They have no place in careful writing or a formal message.
In short: 啦 announces something new, 嘛 recalls something already known or obvious.
Examples
车修好啦！
Chē xiū hǎo la!The car is fixed!
下雪啦！快出来看！
Xià xuě la! Kuài chūlái kàn!It's snowing! Come and look!
好啦好啦，别生气了，我请客。
Hǎo la hǎo la, bié shēngqì le, wǒ qǐngkè.All right, all right, don't be angry, my treat.
你想吃就吃嘛，别客气。
Nǐ xiǎng chī jiù chī ma, bié kèqi.Eat if you want to, come on, don't stand on ceremony.
这道题这么简单，一看就明白嘛！
Zhè dào tí zhème jiǎndān, yí kàn jiù míngbai ma!This question is so simple: it's obvious at a glance!
小孩子嘛，别生他的气。
Xiǎo háizi ma, bié shēng tā de qì.He's only a child, don't be cross with him.
Common mistakes
✗ 你吃饭啦吗？ → ✓ 你吃饭了吗？ (啦 already closes the sentence, it is not combined with 吗).
✗ 他在哪儿嘛？ → ✓ 他在哪儿呢？ (嘛 never asks a question: it states the obvious or coaxes).
✗ 我们到嘛！ → ✓ 我们到啦！ (to announce something new use 啦; 嘛 would say "it's obvious").