As we know will is generally used in Simple present tense. Eg- Sandy will play well with her friends tomorrow.
Would on the other hand is generally used in hypothetical situations. Eg. Had Jonathan not fallen down the stairs, he would be playing in the game tonight.
Will is generally used for:-
1- 100% certainty Joe will arrive at 3.00
2- Willingness I will pay for lunch
3- Polite request Will you please come too?
Would is generally used for:-
1- Desire I would like to come too
2- Polite request Would you like to come too
3- Repeated past action When I worked there, I would get free meals
Further Note that the conditional tense is formed by combining would with the base form if the verb: would provide. This construction expresses the future from the point of view of the past.
The typical sequences for these types of sentences are either Present + Future or Past + Conditional.
Eg- The scientist BELIEVES that the machine WILL BE wonderful.
And
Eg. The Scientist BELIEVED That the machine WOULD be wonderful.
Illustration:- Last Monday, Mary realized the she will have to spend all of that night rewriting her application because she did not back up her files.
The above sentence is incorrect due to following reasons. Will have (simple future tense) should be would have (conditional tense). Mary made her realization on Monday. At that time, her sleepless night spent rewriting the application was in the future. However last Monday night is now in the past. An action that was in the future (relative to the time of the main verb, realized) but now in the past, must be rendered in the conditional tense. This tense is formed by replacing will with would.
Did not back up (simple past tense) should be had not backed up (past perfect tense).
Hence the correct sentence should be:-
Eg- Last Monday, Mary realized the she would have to spend all of that night rewriting her application because she had not backed up her files.
Most important Difference comes in the if----- then construction.
Particular case (in the future) with certainty,
If Sophie eats pizza tomorrow, Then She will become ill.
If present, Then future.
Another possibility for the particular case (in the present) is the present perfect: If Sophie HAS EATEN pizza, then she WILL BECOME ill.
Unlikely case (in the future)
IF Sophie ATE pizza tomorrow, THEN she WOULD BECOME ILL.
IF hypothetical Subjunctive, THEN conditional.
Case that Never Happened (in the Past).
IF Sophie HAD EATEN pizza yesterday, THEN she WOULD HAVE Become ill.
IF past perfect, THEN conditional perfect.
To form the conditional perfect, use would have+ past participle.
In this sentence Sophie is unlikely to eat pizza tomorrow. The conditional tense (would) shows the hypothetical result of an unlikely or untrue condition.
That took a long time to type. Hope this helps. Cheers.