Transformer-based pretrained models like BERT, GPT-2 and T5 have been finetuned for a large number of natural language processing (NLP) tasks, and have been shown to be very effective. However, while finetuning, what changes across layers in these models with respect to pretrained checkpoints is under-studied. Further, how robust are these models to perturbations in input text? Does the robustness vary depending on the NLP task for which the models have been finetuned? While there exists some work on studying robustness of BERT finetuned for a few NLP tasks, there is no rigorous study which compares this robustness across encoder only, decoder only and encoder-decoder models. In this paper, we characterize changes between pretrained and finetuned language model representations across layers using two metrics: CKA and STIR. Further, we study the robustness of three language models (BERT, GPT-2 and T5) with eight different text perturbations on classification tasks from General Language Understanding Evaluation (GLUE) benchmark, and generation tasks like summarization, free-form generation and question generation. GPT-2 representations are more robust than BERT and T5 across multiple types of input perturbation. Although models exhibit good robustness broadly, dropping nouns, verbs or changing characters are the most impactful. Overall, this study provides valuable insights into perturbation-specific weaknesses of popular Transformer-based models which should be kept in mind when passing inputs.