請幫我說明一下這兩句英文的差別原因。

這是個有趣的問題
這種老外常用的語法, 用文法的角度不是很容易解釋. 個人的說法供您參考.
分詞放在句尾, 我把它當"副詞"看.

副詞多半是直接說明飾動詞的特性, 但是有時候我們要說明"主詞以何種狀態進行(或發生)此動作", 可以用分詞來表達. 用另外的例子說明:
The girl came into the room smiling.
這女孩"微笑著"走進房間.
(The girl smiled while she was coming into the room.)

但是如果寫成:
The smiling girl came into the room.
(The girl who was smiling came into the room.)
這個面帶微笑的女孩走進房間.
[smiling]是"形容詞", 單純描述一個名詞的特性. 相當於一個形容詞子句.
有時放名詞後面:
The girl smiling brilliantly came into the room.
(the girl who was smiling brilliantly came into the room.)

從這個角度看, 版主的兩個例子其實是屬於相同的結構.
[is]在這裡是一個完全不及物動詞, 和[He is a boy.]裡的[is](不完全不及物, 需要補語[a boy])性質不同.
There is a bird in the tree singing.
是下面這句的倒裝型態:
A bird is(存在) [there][in the tree][singing].
同樣地:
There is a dog sleeping in the living room. =>
A dog is [there] [sleeping][in the living room].

我們知道, 副詞的位置是相對比較自由的.
[there][in the tree][singing]這三組副詞都是修飾動詞[is]. 其排列的順序並沒有嚴格的限制. [there] [sleeping][in the living room]這三組副詞也一樣. 因此我說兩個例子其實是屬於相同的結構.

如果寫成:
There is a singing bird in the tree.
[singing]是"形容詞", 單純描述一個名詞的特性. 相當於一個形容詞子句.
There is a bird which is singing in the tree.

kevin 老師是最強的,,

in the tree / in the living room.=< 地方 > 可以放在 1.句首 2.句中 3.句尾
多看些例句就會懂:
There is a bird in the tree singing.
= There is a bird singing in the tree .
= In the tree ,there is a bird singing.