The point of randomness is to gather entropy.
Let's take an example:
I choose my password from the 4 first digits of the website plus the character ! and a number consisting of the ASCII number of the first letter.
Example: PayPal -> payp!70 ; amazon -> amaz!65
The entropy of this password is 0. Because the method is always supposed to be known, and nothing is random in this scheme.
If you use alternatively two passwords, the entropy is 1 bit. (2 choices possible)
On the contrary, if I use a 7 generated letter password with random selection, the entropy raises to log2(267) ≃ 32.9
(26 characters possible for each position then reported in term of bits)
The more you add possible characters (that are random) the more the entropy raises. Each "rule-driven" selection of letters do not add entropy (so for same length passwords, entropy is lower).
So in theory this second password has more strength than the previous with non alphanumeric characters. The problem of password is always the same : remembering the password. This means you need to lower the entropy for yourself. Such algorithm as the first presented above enable your to remember a complex password. The "security" given by this approach is a bet over the knowledge of this algorithm (also called "security by obscurity"). This is considered a bad security practice: a targeted attack on yourself (with knowledge of your scheme) will succeed. But usually it's a concession you make for practical use.
Making sure password uses alphanumeric characters + cases changes + non alphanum charcaters + ... is a practice driven by the goal of making a password with great entropy for brute forces attacks, i.e. attacks which consider no particular scheme in password generation. But for each type of attack your password would have different entropy, possibly 0.
To conclude, entropy stays a theoretical measure of password strength, in real life crackers will use a given method based on assumptions. So even if a truly random 6 characters password is consider equally secure if it contain only lowercases letter or both, in practice crackers may choose a method that will lower it.