As Zvoni said: never use data/time conversion routines, at least, not without giving it an explicit format string!
I noticed you have using Zeos components and declared TDateField fields. Since sqlite doesn't have corresponding types for date/time, I can only speculate how Zeos converts them internally. It may even use DateToStr() as you do. Anyway, you can never be sure with, unless you unify all your conversions to a known representation. ISO-8601 is OK since it is lexicographic comparable at DB level.
So, you can replace your TDateField fields with TStringField and convert them to/from ISO-8601 on each access.
You may choose other field type - TLargeintField for unix time for example, or TFloatField for directly storing the TDateTime values. Then processing will be much easier, but with the disadvantage that you can't easily read them by eye.
And don't use LIKE operator, it's not meant for that.