In solid-phase peptide synthesis, why are the formation of aspartimides a problem but not "glutamides"?
Since you understand how aspartimides (succinimides) form and why are they a problem in solid-phase peptide synthesis (SPPS), I'm not going to elaborate it further. Your main question is why don't glutamic acid residues face the same problem? Then, my question is who said they (glutamic acid residues) won't? Of course, they do. One example is discussed in Ref.1. You didn't recognized this cycle formation because the cyclic form is called Glutarimide Formation (contrast to aspartimides, which is actually a succinimide). The abstract states that:

[...] We found that the sequence of Glu(Gly)-OAll leads to glutarimide formation, which disrupts the formation of cyclic RGD peptides. However, our results show that glutarimide formation is sequence dependent and can be inhibited by incorporating an amino acid like Lys(Boc) with steric hindrance from the protecting group. To prevent glutarimide formation, Ser(tBu) was used to replace the glycine in the GGG spacer adjacent to the residue of Glu, and a tailed cyclic RGD peptide, c[RGDfE(SGGKK-NH2)] was successfully obtained.


Initially, $\ce{c[RGDfE(GGGKK-NH2)]}$ was designed with $\ce{c(RGDfE)}$ linked through $\ce{Glu}$ to a tail consisting of a spacer of three $\ce{Gly}$ residues $\ce{(GGG)}$ and a linker of two $\ce{Lys}$ residues $\ce{(KK)}$. We found that the sequence of $\ce{Glu(Gly)-OAll}$ leads to glutarimide formation, which disrupts the formation of cyclic RGD peptides (Note: $\ce{Glu(Gly)-OAll}$ is glutamic acid(glusine) $\alpha$-allyl ester):

The succinimide and glutarimide formations are not just common in SPPS. They can be formed under physiological conditions and can disrupt the biological processes (e.g., Ref.2,3).
References:

Junmin Zhu and Roger E. Marchant, &quot;Solid-Phase Synthesis of Tailed Cyclic RGD Peptides Using Glutamic Acid: Unexpected Glutarimide Formation,&quot; J. Peptide Sci. 2008, 14(6), 690–696 (DOI: https://doi.org/10.1002/psc.975).
Josipa GrzeticJosipa Grzetic and Jos Oomens, “Spectroscopic Identification of Cyclic Imide b2-Ions from Peptides Containing Gln and Asn Residues,” J. Am. Soc. Mass Spectrom. 2013, 24(8), 1228–1241 (DOI: https://doi.org/10.1007/s13361-013-0661-6).
Josipa GrzeticJosipa Grzetic and Jos Oomens, “Effect of the Asn side chain on the dissociation of deprotonated peptides elucidated by IRMPD spectroscopy,” International Journal of Mass Spectrometry 2013, 354-355, 70-77 (DOI: https://doi.org/10.1016/j.ijms.2013.05.005).