Does it matter which end of a chain one sets to be 1 when drawing the structure of organic compounds?
Both of those are the same molecule (flip the one on the left over its horizontal axis, and they are identical).  It does matter where you start counting from, because the numbers chosen for the functional groups should be as low as possible:

Numbering of the chain. This is done by first numbering the chain in both directions (left to right and right to left), and then choosing the numbering which follows these rules, in order of precedence:

Has the lowest-numbered locant (or locants) for the suffix functional group. Locants are the numbers on the carbons to which the substituent is directly attached.
Has the lowest-numbered locants for multiple bonds (The locant of a multiple bond is the number of the adjacent carbon with a lower number).
Has the lowest-numbered locants for double bonds.
Has the lowest-numbered locants for prefixes.


 From IUPAC nomenclature 
So, given the choice between 1,3,4-trimethylcyclohexane and 1,2,4-trimethylcyclohexane, you would choose the latter (for both molecules).