Present tense: My dog will not let us catch him. [when he runs off]. A generality.
Past tense: My dog would not let us catch him.
[when he ran off yesterday] [past tense of will: would, a generality in the past]
Past tense: My dog didn't let us catch him. [last time we played, for example]
They are just two different sentences.
Here, would can be the past tense of will.
When to use will or would The word will is an auxiliary verb (helping
verb) and a modal verb. It is commonly used together with other verbs
to express expectations, determinations, capabilities, and several
other meanings. For example:
My pug will eat anything if you let her. The forecast said it will
rain tomorrow.
Typically, a snake will shed its skin several times
during its life.
Will is also used with every verb to form the future
tense, as in I will go to the beach tomorrow.
The word would is the past tense form of will. In general, it is used
in all the same senses of will to express that something occurred in
the past. habitually.
For example:
My grandparents claimed they would walk ten miles to get to school.
When I worked as a mascot, kids would always be happy to see me.
During her time as a food critic, she would visit many fancy
restaurants.
would versus will