I can see your dilemma looking at the two tone caps. The caps are hygrophanous, which means that they change colour as they dry out. In this case they are drying out from the centre, which turns paler first, giving the two coloured effect. Both species you mention do typically dry out in a similar manner.
If you look at some images that show the stems of
Kuehneromyces mutabilis, Sheathed Woodtuft, you will see that it is very brown in colour and it is scaly up to a ring zone, although this can wash away in the rain. This is the sheathing referred to in the common name. The stem is also quite tough.
Psathyrella piluliformis,
Common Stump Brittlestem, on the other hand has a whitish and fairly fragile stem (as can be seen in your photo).
If you took a spore print you would also find that the
Psathyrella has a much darker spore colour - dark purple brown compared to the more rusty brown spore print of the other species.
Ken