Your "random strings" are what cryptographers would call keys: a key is a secret value with no implied internal structure (as opposed to a password which is "a key which fits in a human brain").
In your first case, users will be granted access based on their ability to present a specific sequence of 10 characters in the a-zA-Z0-9 alphabet. These are 62 possible signs, so there are 6210 = 839299365868340224 possible keys. An attacker trying to force his entry in will have to "guess" the sequence, which he cannot do except by trying possible keys in any order (we assumed "no implied internal structure", which means "uniform probability"). On average, the attacker will have to try half of the possible keys before succeeding, and that's 419649682934170112.
Now suppose that we are in the second case: the user will be accepted only if he enters two 5-character strings in two text fields. Both values are to be right if access is to be granted. In that situation, there are 625 combinations for each string, for a grand total of 625*625 = 839299365868340224 possibilities, which is exactly the same count as previously, so it does not matter.
That is, unless you botch it. "Botching it" would mean, here, telling the user if he got one of the two 5-character strings correct, even if the other is not. For instance, the two entry fields are in successive pages, and your server will show the second page only if the first string was correct. In that case, the attacker can crack both 5-character strings one after the other: that's 625/2 = 458066416 tries on average for the first string, then 458066416 again for the second string. The attacker then succeeds in (on average) 916132832 tries, i.e. 458066416 faster than with the non-botched implementation. That's what I call an epic fail. (Such a failure has already happened, from Microsoft themselves.)
It is simpler, and safer, to require entry of all 10 characters in one go. How you display them, as one text field, two text fields, or a bunch of heart-shaped balloons floating in the screen, has no importance, as long as you do not leak information about which ones are correct. The only information that the verification system should give is whether all characters were correct, or at least one was not.