Students of either Portuguese or Spanish know that that the two languages are related. Each is derived from Latin with a strong Arabic component due to the 800 plus years of Arabic presence in the Iberian Peninsula.
Portuguese, however, has an influence from French because its earliest monarchs were French, which explains the nasal component in Portuguese, but not in Spanish. Both languages, of course, derive their standard versions from the languages of their respective courts. Both also evolved in the centuries previous to the modern era when travel was far more difficult than it is now and when communities were consequently more isolated than at present, and hence, less subject to the influences of one another.
Speakers of both languages are aware of the fact that the non-Portuguese speaking Spanish speaker understands less Portuguese than the converse situation, in which non-Spanish-speaking Portuguese speakers understand more Spanish than their Spanish counterparts. This is partly because Portuguese has conserved more ancient usages than has Spanish. For instance, Portuguese features a robust future subjunctive, a tense that only exists historically in Spanish. It's evident in texts from centuries past, but most Spanish speakers are unaware of it. That is, when the Portuguese speaker says "I will go if you go", the second clause is expressed in the future, whereas in modern Spanish the second clause is expressed in the present tense. Spanish speakers regularly confuse the Portuguese future subjunctive with the imperfect subjunctive. This is also true of the simple form of Portuguese's pluperfect, which is used in texts.
Another feature of Portuguese is the personal infinitive developed in Brazil. This useful personalizing suffix appears in verb plurals and consists of adding a first person nos or em, corresponding to "we" and "they" to the verb. It's as if, hypothetically, one could say "If they go-they" or "If we go-we", something that strikes the Spanish speakers as simply unnecessary.
Spanish speakers are apt to confuse the personal infinitive with a tense, which it is not. You might say that Portuguese is more emotive than Spanish, which the Portuguese speaker may consider slightly on the histrionic side.
As a matter of fact, Portuguese speakers also regularly substitute subject pronouns for object pronouns in good-natured and very wide-spread violation of the rules of grammar. We're not sure if this has to do with the generally more tolerant, laid back and even humorous view of life that is so notable in the Portuguese as contrasted with the Spanish temperament. This controversial point, which is interesting but beyond the scope of this essay, is explained in an amusing essay by Luis Fernando Veríssimo called "Touradas", in which he uses the bullfight as an example of a putative Portuguese tolerance (the bull gets mad but does not die), in contrast to the inevitable Spanish treatment of the bull. Veríssimo extends the analogy to other differences between the two populations. He believes that Portuguese-speaking dictators are awful, but the Hispanic dictators are, well, worse.
The Portuguese language is phonetically far richer than its beautiful, more staccato, neighbor to the east. Nasals, as well as open and closed sounds, are, in fact, meaning changers. To the uninitiated Spanish speaker, these sounds are unperceived as grammatical and meaning-laden functions and, consequently, they have to be consciously learned.
So, while the Spanish speaker may confuse wood (pau) with bread (pão) or grandma (avó) with grandpa (avô), the Portuguese speaker has no such difficulty in understanding Spanish. This is why Spanish speakers often misinterpret Portuguese, and assume that the Portuguese speaker is speaking in the present, for instance when a Portuguese speaker is using the future subjunctive, or that the future subjunctive is in fact similar to the imperfect subjunctive of Spanish. In the case of the phonetic meaning and tense changers, the Spanish speaker has to consciously learn a series of lessons, whereas there is no equivalent challenge for the Portuguese speaker trying to understand Spanish.
Curiously, there is a big difference in the use of reflexive and reciprocal-type pronouns with which Spanish is so liberally sprinkled. Portuguese speakers have no such preference. That is, when the Spanish speaker says "I'm going to have myself a hamburger" the phrase is not colloquial. Portuguese speakers use reflexive and reciprocal pronouns far less, as in English.
If you listen with a musician's ear you might say that Portuguese is a pipe organ and Spanish, a piano or harpsichord. Spanish is contained within and overlapped by Portuguese. However, since the opposite is not the case, Spanish speakers may wonder why their Lusophone interlocutors seem to be having an easier time with communication.