What are the effects of the common cold in an immunodeficient person?
In one small study, rhinovirus in the immunocompromised led to significant mortality from lower respiratory infection:


  Among high-risk patients with cancer, rhinovirus infections are often fatal. In a study of 22 immunocompromised blood and marrow transplant recipients who were hospitalized with rhinovirus infections, 7 (32%) developed fatal pneumonia. The remaining patients had infections confined to the upper respiratory tract. In 6 of the 7 fatal cases, rhinovirus had been isolated in bronchoalveolar lavage fluid or an endotracheal aspirate before death.


Note that this is in hospitalized patients; it says nothing of non-hospitalized patients.

This conclusion has been disputed.

In a slightly larger study among people with hematological cancers (presumed immunosuppressed because of same):


  Respiratory viral pathogens are a common cause of morbidity in patients with hematologic malignancies. ...Both a rapid viral culture with direct fluorescence antibody (DFA) staining and a PCR-based assay (MultiCode-PLx Respiratory Virus Panel) were performed on patients with hematologic malignancies, who underwent collection of a nasopharyngeal swab or bronchoalveolar lavage from October 2006 to April 2007. Eighty-two samples from 70 patients were obtained; all patients had upper respiratory tract symptoms. Respiratory viruses were detected in 10 samples (12%) by conventional virological methods and in 31 samples (38%) by the MultiCode-PLx assay. ...40% of these patients had pneumonia in addition to the upper respiratory tract symptoms. [emphasis mine; note, there is no mention of mortality.]


So long story short, they have stuffy, runny noses, sore throat, cough, etc. Clearly the virus itself causes damage to the mucosa; that is integral to viral replication After entering a mucosal cell, the virus replicates, then the progeny virus is released by lysis of the cell. This damage itself causes inflammation (not the same as an immune reaction), pain, etc. The major difference seems to be a more severe and prolonged experience.

Respiratory Consequences of Rhinovirus Infection
Detection of respiratory viruses with a multiplex polymerase chain reaction assay (MultiCode-PLx Respiratory Virus Panel) in patients with hematologic malignancies