It is a very much philosophical difference, because, depending on the specifics, they are both the same.
The big differences are:
Whom do they benefit?
Very often tax breaks are cut out to benefit the wealthy. They don't have to, but often they do. That is especially true if the tax break is in some form progressive (e.g. cut x% of the tax, or if it is a fixed amount that is deducted before tax in a progressive system). It is also true if it is a non-refundable tax break, because those who pay no taxes don't benefit from them.
Stimulus checks, if distributed to everyone equally, usually benefit the poor more, because €1000 extra might double a poor person's monthly income, while a wealthy person might not even notice it.
When are they paid?
Generally speaking, a tax break is paid after the fact (e.g. at the end of the year) while stimulus checks are paid in advance.
Again, since tax breaks are paid after the fact, they are less helpful for the poor, since they might not have enough money to wait until the end of the year.
How often do they get dispensed?
Tax breaks tend to be a regular thing, while stimulus checks are usually one-time payments.
... it is totally possible for either of those to be used for the opposite purpose. E.g. in Austria we had a so-called Family Tax Break which would give families a €1000 refundable tax break that is paid directly when you get your wages or alternatively (if you don't get wages) at the end of the year. Also, this was limited to one year.
So it primarily benefited the poor, was paid not quite in advance but in a timely manner, and was a one-time payment. So it's closer to the spirit of a stimulus check, even though it was a tax break.
On the other hand, stimulus checks can also be tied to requirements, which then only benefit a certain segment of the population, e.g. business owners. So they don't necessarily have to help poor people either.