How much do you have to respect checklist order?
Yes, the published sequence must be respected. Here is a quote from published instructions on checklist usage by one of the major manufacturers based in the USA:

&quot;. . . . The crew may need to stop a checklist for a short time to do other tasks. If the interruption is short, continue the checklist with
the next step. If a pilot is not sure where the checklist was stopped,
do the checklist from the start. If the checklist is stopped for a
long time, also do the checklist from the start.&quot;

If this philosophy is not already spelt out for some current aircraft types, the Regulatory bodies and Industry are (correctly) pushing it that way. Tedium is not considered as a sufficient reason to dump the C/L sequence, it is actually one reason for having a checklist!
The purpose of cockpit checklists for both Normal and Abnormal/non-Normal situations would largely be defeated if the published sequence were not followed as they are designed to ensure that the aircraft is correctly configured and safe for the next step.
In an unusual situation though (read esp: impending catastrophe), the above is to be understood in the light of the necessary disclaimer, quoted from the same manual that's quoted above:

&quot;The flight crew must be aware that checklists cannot be created for all conceivable situations and are not intended to replace good
judgment. In some situations, at the captain’s discretion, deviation
from a checklist may be needed.&quot;

The statement - “ . . even if it makes no difference . . “ is a tricky one and it evokes a response of, “where do you draw the line?&quot; - checklists are a manifestation of the need for Standardisation.
Here's a sum up, intended to complement the advice given by  many on this page.

Checklists are made very scientifically, but even so, the endeavor is based on a best effort basis and checklist philosophy is constantly evolving.
instances of abnormal/non-normal  situations can exist or can be conceived to support either side of the discussion.
Checklists are aimed at providing the highest probability of a successful outcome based on the best knowledge of the day and age.
Institutional Policy must be sufficiently evolved to look at variations from SOP in a practical manner.
A C/L sequence that regularly invites 'modification’, as reported by crew or as found by a surveillance system, requires a rethink.
In current day practice, ‘one off’ occurrences of a breach of checklist sequence are looked at in a benign manner by all but a few agencies.
As a policy, “not following the checklist sequence” cannot be  regularly condoned.