Modulate means a change of key. Something needs to make clear the music is in a new key. Traditionally a cadence in a new key does the job. Importantly to the comparison of terms a modulation can be achieved with all new thematic material.
In transposition some musical material is repeated, but at a different position pitch-wise. There is both chromatic and diatonic transposing so the pitch shifting isn't necessarily by the exact same interval for each note. Either way the important thing is that some thematic material is repeated. Note the difference with modulation where repeating material isn't part of the definition.
Given that transposition can be diatonic, it won't necessarily result in a key change. (You may notice that a diatonic transposition is the same as a diatonic melodic sequence.)
If a thematic passage was chromatically transposed, it could result in a key change, but I suppose it really depends on how the passage moves, and also the length of the passage. Some might end up resulting in a sequence. This
[V6/5 of IV|IV]|[V6/5 of V|V] is both a transposition of the material in brackets and a harmonic sequence. On the other hand in a simple sonata form the "second theme group" (the part of the exposition in the dominant key) can often be simply transposed down a fifth to the tonic during the recapitulation. (If that sonata stuff isn't familiar, try reading some reviews of sonata form and then look at L. Mozart's Nannerl Notebook which as many examples in small scale pieces.) Transposition in the recapitulation is a nice example where it has formal, structural meaning and a modulation, a key change, is the result.
So, there is some overlap in the meanings, but modulation doesn't really say anything about the handling of thematic material while transposition does tell specifically we are modifying thematic material.