Does re-iterate AES-128 with different keys gives any benefict? I'm not cryptography expert but I have this doubt…

Suppose I have 2 different keys:


secretkey1
secretkey2


And a plain text:


myReallySecretPlainText


If I use AES-128 on my plain text with my first key I get this:

U2FsdGVkX19fHxumDqEB9Kuyvhj+aoHpctLCdo2VEvt1danATMo+sBaz3nsCaS/Z


now, if I use on this cipher text my second key I got this:

U2FsdGVkX1/AqJ3w8nTRAHvm1ZbDTdwmq7LTVBDR/z4O9ngaXGuAIqcqTzPj+flPRIV6++twqBnIEHgUVVe7oO1D75VO14ezRX5tqcjBTPUUtr7YDqCks5ZW7fX8crqg


Looks pretty safe, but I have a doubt: does this have the same security of a AES-256? Or does it have the security equivalent to a single 128 bit key?
The idea you describe is vulnerable to a meet-in-the-middle attack that work in approximately $2^{128}$ time and $2^{128}$ memory. The attack assumes knowledge of plaintext/ciphertext pair(s). Given a pair, you encrypt the plaintext with every possible key 1 and store those values. You then decrypt the ciphertext with every possible key 2 and look for a match between the two datasets. Say you find that $E_{k_1}(P)=D_{k_2}(C)$. Likely these are your two keys. This can be validated by looking at additional plaintext/ciphertext pairs. 

As codesinchaos says, the attack may not necessarily be applicable with realistic cost models, but it is a theoretical break and is therefore considered weaker than just using a larger key space in the first place. Also, I've got to believe that this idea is slower than just using AES-256.