Both are different ways to express the boundary layer thickness in a deterministic way.
The boundary layer according to wikipedia:
a boundary layer is an important concept and refers to the layer of fluid in the immediate vicinity of a bounding surface where the effects of viscosity are significant.
Since there can be debate about what significant means, multiple formulations for the boundary layer thickness exist. And depending on the problem to solve, one of them can be more suitable than the other.
Therefore the boundary layer thickness is by typically specified as the distance where the velocity reaches 99% of the free stream velocity.
The boundary layer momentum thickness can be exactly specified as the distance a uniform flow field should be displaced by to equal the total momentum flux
(m*v)*v of the real boundary layer (non-uniform).
There is also the boundary layer displacement thickness, comparing the mass flux deficit
(m*v) rather than the momentum flux deficit
(m*v)*v, as highlighted by the slides in Juan Jimenez's comment.
Take a look at this quora answer as it contains good explanatory figures.
I found a paper with comparative figures of the displacement and momentum thickness based on XFOIL results. It's worth to take a look at. Iliev, Sergiu. (2016). Aerofoil Analysis using XFOIL - Practical Implementation for Preliminary Wing Design. 10.13140/RG.2.2.30861.54243.
An example of where displacement and momentum thicknesses are useful can be found in the XFOIL documentation.
The displacement thickness can be used to couple a boundary layer solver with a potential flow solver (ISES code).
XFOIL uses the momentum thickness to calculate the viscous drag.
The boundary layers and wake are described with a two-equation lagged
dissipation integral BL formulation and an envelope e^n transition
criterion, both taken from the transonic analysis/design ISES code.
The entire viscous solution (boundary layers and wake) is strongly
interacted with the incompressible potential flow via the surface
transpiration model (the alternative displacement body model is used
in ISES). This permits proper calculation of limited separation regions.
The drag is determined from the wake momentum thickness far downstream.
A special treatment is used for a blunt trailing edge which fairly
accurately accounts for base drag.