What is the logical difference between "to seek" and "to look for"?

I regard them as synonyms. However, there is a lexical difference. 'Seek' is a pure verb and 'look for' is a phrasal verb - a pure verb plus, in this case, a preposition. Phrasal verbs carry an idiomatic meaning and are more typical in spoken or informal usage.

A similar pair might be 'discover' and 'find out' - but we would never think to transfer the preposition from the phrasal verb and use it with the pure verb - 'discover out'.

Perhaps the applicant is using a telephone to seek employment?

This would be considered seeking but not literally looking.

In a non-literal sense, "looking for" is synonymous with "sniffing out" but I don't think you can use that to argue that these terms are synonymous in every other context in which one of them might be used.

Logically, seeking encompasses a greater variety of methods and senses than does looking.